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Final Report
Optimized Development of Urban Transportation
Networks
Paul Schonfeld
Maryland Transportation Institute
1173 Glenn Martin Hall
University of Maryland, College Park
MD 20742
Date
May 11, 2019
Prepared for the Urban Mobility & Equity Center, Morgan State University, CBEIS 327, 1700 E. Coldspring Lane,
Baltimore, MD 21251
2
ACKNOWLEDGMENT
The research presented in this report was partly funded by the Urban
Mobility & Equity Center led by Morgan State University. Its support is
greatly acknowledged.
Disclaimer The contents of this report reflect the views of the authors, who are responsible for the facts
and the accuracy of the information presented herein. This document is disseminated under
the sponsorship of the U.S. Department of Transportation’s University Transportation
Centers Program, in the interest of information exchange. The U.S. Government assumes no
liability for the contents or use thereof.
©Morgan State University, 2019. Non-exclusive rights are retained by the U.S. DOT.
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1. Report No. 2. Government Accession
No.
3. Recipient’s Catalog No.
4. Title and Subtitle
Optimized Development of Urban Transportation Networks 5. Report Date
May 11. 2019
6. Performing Organization Code
7. Author(s) Include ORCID #
Dr. Paul Schonfeld orcid.org/0000-0001-9621-2355 8. Performing Organization Report No.
9. Performing Organization Name and Address
Maryland Transportation Institute
1173 Glenn Martin Hall, University of Maryland College Park,
MD 20742
10. Work Unit No.
11. Contract or Grant No.
69A43551747123
12. Sponsoring Agency Name and Address
US Department of Transportation
Office of the Secretary-Research
UTC Program, RDT-30
1200 New Jersey Ave., SE
Washington, DC 20590
13. Type of Report and Period Covered
Final
14. Sponsoring Agency Code
15. Supplementary Notes
16. Abstract
This report presents improved methods for planning and scheduling interrelated improvements in
transportation networks. Due to the complexity of the relevant evaluation functions, which cannot be
optimized with classical calculus techniques, the proposed methods rely on customized genetic algorithms for
optimizing the selection, sequencing and scheduling of the interrelated alternatives. Three applications to
urban transportation networks are presented in journal papers which are included in appendices. The papers
demonstrate the applicability of the proposed methods to urban road networks, to intersections in urban road
networks and to the development of urban rail transit networks.
17. Key Words :
Network Development, Network Optimization, Project
Scheduling, Interrelated Projects
18. Distribution Statement
19. Security Classif. (of this
report) : Unclassified
20. Security Classif. (of this
page)
Unclassified
21. No. of Pages
68
22. Price
4
Optimized Development of Urban Transportation
Networks
Paul Schonfeld
University of Maryland, College Park
Final Report for the University Mobility and Equity Center
May 11, 2019
5
Table of Contents
Abstract
Executive Summary
Appendix 1 – Selecting and Scheduling Interrelated Projects
Appendix 2 – Selecting and Scheduling Link and Intersection Improvements in Urban
Networks
Appendix 3 – Optimal Development of Rail Transit Networks
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Abstract
This report presents improved methods for planning and scheduling interrelated
improvements in transportation networks. Due to the complexity of the relevant evaluation
functions, which cannot be optimized with classical calculus techniques, the proposed
methods rely on customized genetic algorithms for optimizing the selection, sequencing and
scheduling of the interrelated alternatives. Three applications to urban transportation networks
are presented in journal papers which are included in appendices. The papers demonstrate the
applicability of the proposed methods to urban road networks, to intersections in urban road
networks and to the development of urban rail transit networks.
Executive Summary
In this project methods were developed for planning, evaluating and scheduling
improvements in transportation networks in order to optimize the development of such
networks in response to evolving demand and societal objectives. The work was performed at
the University of Maryland, College Park, in the years 2017 and 2018, with funding from the
University Center for Mobility and Equity led by Morgan State University, as well as from
other sources. The work was directed by Professor Paul Schonfeld from the University of
Maryland’s Department of Civil and Environmental Engineering. Important contributors
included his students Elham Shayanfar, Uros Jovanovic and Ya-Ting Peng, and Professor Zi-
Chun Li from the Huazhong University of Science and Technology in Wuhan, China.
The problems of selecting and scheduling improvements in transportation networks are
greatly complicated by the pervasive interrelations among candidate alternatives. In
engineering economics and other fields the alternatives are classified as (a) mutually
exclusive, (b) independent, and (c) interrelated. The alternatives are considered mutually
exclusive if only one alternative may be chosen, the others being necessarily rejected. They
are independent if the benefits and costs of each alternative do not depend on which other
alternatives are selected or when the other alternatives are implemented. If the benefits or
costs of alternatives depend on which others are selected and when all are implemented, the
alternatives are classified as interrelated. While generally accepted methods for analyzing
mutually exclusive and independent alternatives can be found in standard textbooks no such
general methods are found for analyzing interrelated alternatives. Furthermore, even the
methods that have been designed for analyzing interrelated alternatives in some specific
applications have been deficient in their abilities to deal with complex interrelations,
dissimilar types of alternatives, multiple uncertainties, scheduling decisions, realistic problem
sizes and other important factors.
The deficiencies of methods for analyzing interrelated alternatives constitute a major gap
in the state of the art in engineering economics, operations research, and related fields. This is
especially unfortunate since interrelated alternatives pervade the world. For example, in
transportation systems, which are the primary focus of our proposed study, improvements to a
network’s various links and nodes are interrelated partly because such improvements
redistribute flows in networks. Each improved link may divert traffic from parallel links, shift
congestion and capacity bottlenecks to other links in-series, reduce the need for other
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improvements, and thus affect the benefits obtainable from improving other network
components. Hence the benefits of various improvements may add up non-linearly. Some
improvement projects may be synergistic while others may be largely wasted or even
counterproductive (e.g., according to the Braess Paradox) when combined with other
improvements.
Beyond interrelations due to non-linearly additive benefits (including some externalities),
alternatives may be interrelated through their costs (e.g., through economies of jointly
constructing several projects), their budget constraints and other financial relations, their
constructability or operability requirements, political or equity considerations, and in other
ways.
In addition, decisions regarding infrastructure maintenance or development are subject to
substantial uncertainties regarding future demand or usage, costs, finances, implementation
schedules, and future component performance (including capacity, delay, deterioration, and
failures). Methods have been developed for dealing with uncertainties in capacity expansion
and maintenance for infrastructure projects but these are far from adequate in dealing with
realistic numbers of interrelated projects and their applicability is limited.
The three appendices of this report present three papers on the analysis of interrelated
alternatives for transportation networks. The first paper (in Appendix 1) by E. Shayanfar and
P. Schonfeld, entitled “Selecting and Scheduling Interrelated Projects: Application in Urban
Road Network Investment,” presents a metaheuristic method based on a genetic algorithm for
optimizing network development problem. The metaheuristic approach is needed because for
realistic problem sizes the objective function is very unsmooth and not solvable with either
classical methods of mathematical analysis or with mathematical programming approaches.
The paper shows how a genetic algorithm can be formulated and applied to efficiently solve
this problem. In effect, the method consists of expressing all possible sequences for
implementing alternatives as genetic chromosomes, translating the sequences into exact
development schedules (in continuous time rather than discrete periods) by applying the
binding constraints (which, in this case, are the budget constraints) and using a relatively
simple traffic assignment algorithm to estimate traffic speeds and volumes throughout a
multi-year analysis period for any development schedule. The traffic speeds and volumes can
then be used to estimate other effectiveness measures, including travel times and user costs,
throughout the analysis period.
Since heuristic methods do not guarantee that a global optimum is always found, the paper
shows how a statistical test can be used to confirm that the infinitesimal probability of finding
significantly better solutions than those found by the proposed heuristic method. Thus, it can
be demonstrated that any errors due to the proposed algorithm are negligible compared to
unavoidable errors in estimating inputs regarding the actual transportation system and its
future demand characteristics.
The second paper (in Appendix 2) by U. Jovanovic, E. Shayanfar and P. Schonfeld, titled
“Selecting and Scheduling Link and Intersection Improvements in Urban Networks“ shows
how the analysis, selection and scheduling of interrelated components in urban road networks
8
can be extended to include improvements at intersections, i.e., widening the intersections with
additional lanes through them. To accomplish this, the traffic assignment model had to be
adapted to analyze intersection flows and delays. This was accomplished by introducing into
the previously used Frank Wolfe assignment algorithm pseudo-links for each turning and
through movement at each intersection, e.g., 12 pseudo-links at each full four-leg intersection.
Delays on the pseudo-links were estimated with a model developed by Akcelik.
The third paper (in Appendix 3) by Y. T. Peng, Z.C. Li and P. Schonfeld, titled “Optimal
Development of Rail Transit Networks over Multiple Periods,” shows how the analysis,
selection and scheduling of interrelated network components can be extended to optimize the
phased development of a rail transit network, In this problem it is assumed that the locations
of rail lines and stations in the network are pre-determined. The remaining decisions are about
which links and stations should be added at what time, depending mainly on demand growth,
available external budgets and usable fare revenues from network segments that are already
operating.
The methods developed and tested in this project are already usable for evaluating selecting
and scheduling interrelated network improvement projects. Beyond the accomplishments of
this project, desirable improvements would include improved consideration of uncertainties
(e.g. in demand, costs, budgets and construction times) and extensions to multi-modal
transportation systems.
9
Appendix 1 - Shayanfar, E. and Schonfeld, P. “Selecting and Scheduling
Interrelated Projects: Application in Urban Road Network Investment,”
International J. of Logistics Systems and Management, 29-4, 2018, pages 436-454.
Selecting and Scheduling Interrelated Projects:
Application in Urban Road Network Investment
Elham Shayanfar and Paul Schonfeld
Accepted for International Journal of Logistics Systems and Management, Oct. 2016
ABSTRACT
Decisions about the selection of projects, alternatives, investments, operating policies and
their implementation schedules are major subjects in various fields including operations
research, financial analysis, business management, engineering economy and transportation
planning. In these various disciplines sufficiently good methods have been developed for
planning and prioritizing projects when interrelations among those projects are negligible.
However, methods for analyzing interrelated alternatives are still inadequate. We propose a
combinatorial method for evaluating and scheduling interrelated road network projects.
Specifically, this paper demonstrates how a traffic assignment model can be combined
effectively with a Genetic Algorithm (GA) in a multi-period analysis to select and schedule
road network projects while capturing interactions among those projects. The goal is to
determine which projects should be selected and when they should be funded in order to
minimize the present value of total system cost over a planning horizon, subject to budget
flow constraints.
KEYWORDS: Project selection and scheduling, Genetic Algorithm, GA, Project
interrelations, User equilibrium, Project evaluation, System optimization, Planning and
prioritizing projects, Minimizing system cost
10
1. INTRODUCTION Evaluating transportation infrastructure projects and determining which should be
implemented at what time has been the subject of ongoing studies for decades. Commonly
used evaluation practices aggregate linearly the project impacts in the objective function,
which is then optimized. Such practices are often inadequate, especially for projects in
transportation networks, since they disregard possible interrelations among projects due to
non-linearly additive benefits, costs, budget constraints, constructability or operability
requirements, and other possible factors. This paper deals with road expansion projects as an
example of interrelated projects. However, the method proposed here for project selection and
scheduling may be used to analyze interrelated alternatives in general cases if methods for
evaluating objective functions are available.
In various disciplines sufficiently good methods have been developed for dealing with
projects which are not interrelated. In general, alternatives are classified as (a) mutually
exclusive, (b) independent and (c) interdependent or “interrelated”. The alternatives are
considered mutually exclusive whenever implementing one project automatically excludes the
others. Alternatives are independent if their benefits and costs do not depend on which other
alternatives are selected or when the other alternatives are implemented. Otherwise, the
alternatives are classified as interdependent. Although generally accepted methods for
analyzing mutually exclusive and independent alternatives are available in the literature, no
such general methods are found for analyzing interrelated alternatives. Even the methods that
have been designed for analyzing interrelated alternatives in some specific applications have
been incapable of dealing with enough interrelations and realistic problem features.
The problem of evaluating and selecting interdependent alternatives exists in various
fields including economics, operations research, business, management, transportation and
portfolio management. In portfolio management, interrelations between choices (stocks) were
identified and modelled as early as the 1950s in pioneering work by Markowitz (1952). Since
then more recent studies have addressed the problem of portfolio selection among
interdependent projects (Cruz et al., 2014; Li et al., 2016). However, the literature review
shows both insufficient studies on this problem and lack of comprehensive applicable
methods for real world problems especially in the field of transportation.
This study demonstrates how a relatively simple method, namely a traffic assignment
algorithm, can be efficiently used to evaluate the objective function of an investment planning
optimization problem and thereby compute the relevant interrelations among many projects
that are implemented at various times. However, more complex methods for evaluating the
objective functions, such as microscopic simulations, can also be combined with the Genetic
Algorithm (GA) used here for optimizing the project selection and schedule. In recent years,
meta-heuristics have been widely used for finding optimal or near-optimal solutions. The
work presented in this paper is an extension of a previous study conducted by Shayanfar et al.
(2016). That study applied three meta-heuristic algorithms including a GA, Simulated
Annealing (SA) and, Tabu Search (TS) in seeking efficient and consistent solutions to the
selection and scheduling problem. Its main contribution was to compare three meta-heuristics
for this problem in terms of solution quality, computation time and consistency. The
comparative analysis was especially useful in determining which algorithm was preferable in
various circumstances. In summary, the results indicated that the GA yielded a better (lower
total cost) solution than the other two algorithms and yielded the most consistent solutions
(i.e. with the lowest coefficient of variation), indicating that different replications of the GA
yield almost similar final solutions after sufficient iterations.
Therefore, the current paper incorporates the GA used in Shayanfar et al. (2016) while
enhancing its assumptions and contributing to the literature in several ways. First, we
demonstrate how a traffic assignment model can be combined effectively with a GA for
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planning and prioritizing purposes while capturing more interactions among projects, i.e.
beyond the previously considered pairwise interactions. Second, we modify the algorithms’
assumptions to account for the possibility that candidate projects may become economically
justified or unjustified after the implementation of previous projects. This may occur due to
project interrelations and the possibility that the cost savings from completing a project are
affected by earlier project implementations. Third, a multi-period analysis is incorporated in
this study to distinguish between peak and off-peak traffic flows. Fourth, the budget
constraint is reformulated to include possible internal funding from fuel taxes. Fifth, we
assume that the demand changes over time during the planning horizon (growing
exponentially in our example). Finally, we demonstrate this methodology on two example
networks and present a statistical test of the goodness of the heuristic results. Generally, the
methodology presented in this work should also be applicable to other prioritization problems
with interrelated alternatives, which abound in transportation and other activities.
2. LTERATURE REVIEW In engineering economics, a number of studies have developed methods to address the
problem of project scheduling. Beenhakker and Narayanan (1975) formulated the scheduling
problem as a simple integer program assuming projects are independent. The formulation
maximized the total net benefit of all projects subject to a budget constraint. Chiu and Park
(1998) proposed a capital budgeting model under uncertainty in which cash flow information
was considered as a special type of fuzzy number. To prioritize fuzzy projects based on the
present worth of each fuzzy project cash flow, a branch and bound procedure was suggested.
Koc et al. (2009) proposed a model that forms an optimal priority list of projects,
incorporating multiple scenarios for input parameters. For this purpose, a greedy heuristic
algorithm was developed to create the prioritize list. Our research indicates that in the field of
engineering economics and capital investment planning, the methods developed for selecting
and scheduling do not adequately deal with possible interrelations among alternatives.
One of the first works we could find that considered interdependent alternatives was
that of Markowitz (1952) on portfolio management. This study formulated a multi-objective
function minimizing the sum of purchase cost and risks. In this case, a “dependence matrix”
which captures two-way, three-way or n-way interrelations was introduced to model the
interdependence among a set of choices. This method and its variants can also be found in
more recent works. Dickinson et al. (2001) developed a model to optimize a portfolio of
development improvement projects for the Boing Company. The authors used a dependence
matrix to quantify the interdependencies among projects. Then a non-linear, integer program
model was developed to optimize the project selection. Sandhu (2006) introduced a
dependency structure matrix that captured the project logistic interdependencies. Durango-
Cohen and Sarutipand (2007) formulated a quadratic programming for optimizing
maintenance and repair (M&R) policies for transportation infrastructure systems. The
quadratic objective of their work included the pairwise economic dependencies capturing the
costs and benefits of improving adjacent facilities. Bhattacharyya et al. (2011) also considered
n-way interdependencies in the Research and Development (R&D) project portfolio selection
problem.
Two main issues arise from using a dependence matrix. First, as Disatnik and
Benninga (2007) argue, the estimation and manipulation of a dependence matrix becomes
computationally difficult as the project space grows. Second, the pairwise and n-way
12
dependencies do not completely represent the complex interrelations and fall short of the
desired relations among alternatives. Instead of a dependence matrix, complete system
models, such as queueing approximations (Jong and Schonfeld, 2001), equilibrium
assignment (Tao and Schonfeld, 2005), microsimulation (Wang and Schonfeld, 2008) and
neural networks (Bagloee and Tavana, 2012), are better suited for modeling interrelations.
This section reviews the current literature on evaluating and prioritizing interdependent
projects.
The SA algorithm developed by Bouleiman and Lecocq (2003) for the resource-
constrained project scheduling problem aimed to minimize the total project duration. To this
end, they replaced the conventional SA search scheme with a more novel design mindful of
the specificity of the solution space of project scheduling problems. Tao and Schonfeld (2005)
developed a GA to solve the Lagrangian problem, and optimized the selection of
interdependent projects under cost uncertainty. They employed a traffic assignment model to
evaluate the objective value of the Langragian problem and assess the project impacts.
Similarly, Wang and Schonfeld (2005) developed a GA to solve the problem of selecting and
scheduling interrelated lock improvements for a waterway network. They designed a
microscopic waterway simulation model (i.e. which traced every vehicle movement) to assess
the performance of the waterway system while evaluating the project interdependencies.
Dueñas-Osorio et al. (2007) incorporated the interdependence response among network
elements based on geographic proximity i.e. the response of one network given the state of
another network was monitored for various levels of coupling among them. They studied the
network response subject to external and internal disruptions such as attacks, lack of
maintenance and breakdown due to aging. Their work indicated that responses that are
destructive to networks are greater when interdependencies are considered after disruptions.
Tao and Schonfeld (2007) developed island model variants of GA’s for optimizing project
selection and scheduling, and used these models to solve a stochastic optimization problem.
Their work considered how uncertainties in travel times and construction costs affect total
system costs.
Szimba and Rothengatter (2012) developed a framework for integrating the
interdependence among infrastructure projects in classical benefit-cost analysis. They
addressed the complexity of a large-scale interdependence problem by introducing a heuristic
method to optimize the dynamic mixed integer program. In this approach, the number of
projects and their interrelations were reduced stepwise, resulting in a fewer interdependence
cases. They used two procedures to measure the magnitude of interdependencies. In the first,
projects were added to a minimum network configuration. In the second, projects were
deleted from a maximum network configuration. Bagloee and Tavana (2012) used the
Traveling Salesman Problem (TSP) to formulate the prioritization problem. They used a
Neural Network to consider the interdependence among projects, and developed a search
engine influenced by Ant Colony (AC) hybridized with GA to optimize the problem. Li et al.
(2013) developed a multi-commodity minimum cost network (MMCN) to evaluate the impact
of projects, i.e. to estimate the benefits of projects through a life-cycle-cost analysis. They
further proposed a hypergraph knapsack model to maximize these benefits for a set of
interdependent projects. Rebiasz et al. (2014) developed a hybrid method which combined
stochastic simulation with arithmetic on interactive fuzzy numbers and nonlinear
programming. The goal was to solve the problem of capital budgeting, accounting for both
stochastic and economic interdependency between projects.
Chen et al. (2015) reformulated the mixed network design problem (MNDP) to
identify optimal capacity expansions of existing links and new link additions. Their model
was designed to minimize the network cost in terms of the average travel time affected by the
13
expansion of existing links and the addition of new candidate links. In this case a surrogate-
based optimization framework was proposed to solve the MNDP. Bagloee and Asadi (2015)
developed a hybrid heuristic method to optimize the prioritization problem while considering
demand uncertainties. They formulated the objective function as the reduction in users’ travel
time and, introduced a policy based on “gradient maximization” to find solutions. Tofighi and
Naderi (2015) developed a mixed integer linear program to formulate the selection and
scheduling of projects maximizing total expected benefits. They also proposed an ant colony
algorithm to optimize the objective function. This paper defined the interdependencies among
projects with a simple dependence matrix, which is insufficient in capturing the full
interrelations among projects in transportation networks and various other complex systems.
3. PROBLEM FORMULATION Roadway improvement projects are usually interrelated since delays at one link are affected
by operations at other links, both upstream and downstream. Conceptually, if the capacity
increases in one link of a network, congestion and average travel times tend to increase in
other links that are “in series” with it and decrease in its “parallel” links. Therefore, the total
cost saved from multiple projects is not a linear summation of savings from individual links.
Additionally, the interrelation among links is reflected in our budget constraint since the
budget is partly supplied by internal taxes, which may change after each project
implementation, thus complicating this problem.
The objective function for problems such as prioritizing interrelated projects has a
surface that is “noisy” (i.e. containing numerous local optima) and non-convex. Moreover, as
the number of candidate projects increases, the problem’s solution may soon exceed the
capabilities of conventional mathematical optimization methods. Consequently, heuristic
methods have become the preferred approach for solving such problems. In this study a GA is
very useful in effectively finding near-optimal solutions for such a large solution space and
noisy objective function. Our objective function is the net present value of total cost including
both (i) total road user and (ii) total supplier cost subject to budget constraints. The goal is to
specify which links should be selected for expansions in what order, and when they should be
started and completed over the horizon period T.
Therefore, the formulated objective function minimizes the present value of total user
and supplier cost, over a specified planning horizon, subject to a budget flow constraint over
that entire horizon. In this context, the user cost is the total delay for users in the system
multiplied by their value of time. The supplier cost is the present value of implementation
costs for all projects. An additional improvement over some previous studies is the inclusion
of project costs in the objective function. This is necessary since not all selected projects are
guaranteed to fit in the budget and be implemented within the analysis period. In fact, some
projects may be discarded from the sequence as they may become unjustified sometime
during the analysis. Therefore, different solutions (i.e. different sequence of projects) may
entail different project costs which should be considered in the objective function. The
objective function Z to be minimized is the present value of total cost:
𝑚𝑖𝑛 𝑍 = ∑ {𝑣
(1 + 𝑟)𝑗∑ 𝑤𝑖𝑗
𝑛𝑙
𝑖=1
}
𝑇
𝑗=1
+ ∑𝑐𝑖𝑥𝑖(𝑡)
(1 + 𝑟)𝑡
𝑛𝑝
𝑖=1
(1)
14
{𝑥𝑖(𝑡) = 0 𝑖𝑓 𝑡 < 𝑡𝑖
𝑥𝑖(𝑡) = 1 𝑖𝑓 𝑡 > 𝑡𝑖
In this formulation 𝑡𝑖 is the time when project i is completed and ready for use while 𝑥𝑖(𝑡) is a
binary variable specifying whether project i is finished by time t. In the objective function,
𝑤𝑖𝑗 denotes the travel time over link i in year j, and 𝑐𝑖 is the present value of the cost of
project i. 𝑛𝑝, 𝑛𝑙, 𝑣 are the number of projects implemented, total number of links and value of
time, respectively, while r is the interest rate.
In this problem an internal budget source is considered for funding future projects.
Specifically, throughout the analysis period, fuel taxes collected from users are added to an
external budget in determining the overall investment budget. This assumption is realistic, as
fuel taxes and toll collections contribute substantially to highway improvement budgets. The
internal budget is estimated as:
𝑏(𝑡𝑖)𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 = 𝑉𝑀𝑇(𝑡𝑖−1) ∗ 𝑓𝑟 ∗ 𝑓𝑐 ∗ 𝑓𝑡
where 𝑓𝑟, 𝑓𝑐, 𝑓𝑡 denote fuel consumption rate (gal/veicle.mile), fuel cost ($/gal), and gas tax
rate (percentage of tax collected from dollar spent on gas) respectively. This formulation
shows that fuel taxes collected in period 𝑡𝑖−1 contribute to the budget available in period 𝑡𝑖.
𝑉𝑀𝑇(𝑡𝑖−1) presents the vehicle miles travelled during the time project 𝑖 − 1 is completed.
Jong and Schonfeld (2001) formulated the selection and sequencing problem by defining the
decision variables as the completion time of projects. In this formulation the budget constraint
is defined as follows:
∑ 𝑐𝑖𝑥𝑖(𝑡) ≤ ∫ 𝑏(𝑡)𝑑𝑡, 0 ≤ 𝑡 ≤ 𝑇𝑡
0
𝑛𝑝
𝑖=1
(3)
More specifically, under a limited budget, which is continuously distributed over time,
it is efficient to fund and complete projects one at a time, because the system gains immediate
benefits as soon as each additional project is completed and ready for use. The budget
constraint is almost invariably binding because, in actual cases, there are always some
justifiable projects waiting for funding. In fact, funding multiple projects concurrently
increases their completion time, meaning that their benefits are postponed. Therefore,
considering budget limitations, it is preferable to avoid funding overlaps, and fully fund
projects before starting to fund the next ones, and finish each project one at a time. It should
be noted that construction times of projects may overlap even if their budget accumulation
periods do not, if constrained budget flows can be shifted over time (e.g. through lending).
Thus, the optimized schedule of each project is uniquely and easily determined from the
optimized sequence by considering the budget flow.
To date, similar studies have assumed that the set of candidate projects remains
unchanged throughout the analysis period, thus disregarding that due to interrelations,
previous project implementations alter the benefits from completing succeeding projects,
possibly making them economically unjustifiable. It is also possible that initially unjustifiable
projects (i.e. with higher costs than benefits) may become economically desirable, e.g. after
bottlenecks in networks are cleared. Accordingly, in this paper, the undesirable projects (i.e.
15
whose benefits < costs) are temporarily removed from the list of candidate projects, with the
possibility of reentering the sequence after their benefits exceed their costs. In other words,
the set of candidate projects is constantly updated, and acceptable projects may be replaced
unacceptable ones at different stages of analysis.
4. EVALUATION MODEL This paper applies the convex combination algorithm of Frank-Wolf (1956) as an evaluation
model to assess the effects of each expansion project on the network. The Frank–Wolfe
algorithm is an iterative first-order optimization algorithm for constrained convex
optimization widely used for solving traffic assignment problems. In each iteration, the
Frank–Wolfe algorithm considers a linear approximation of the objective function, and moves
slightly towards a minimizer of this linear function. The algorithm starts with an initial flow
x. Subsequently, each iteration performs a direction search by solving a linear approximation
of the objective function which determines the step size and moves in that direction. Finally,
the algorithm terminates when it satisfies a convergence criterion based on the similarity of
successive solutions. In this case, the traffic assignment algorithm provides a relatively simple
model for evaluating solutions (i.e. computing the objective function value), and estimating
link travel times, speeds, volumes, and hence user costs.
5. OPTIMIZATION MODEL In general, simulation methods are reserved for complex problems which are not solvable
analytically. However, it may be computationally expensive to insert simulation modules
directly into optimization loops. Hence, various approximation methods have been substituted
for simulation (Dai and Schonfeld, 1998, Wei and Schonfeld, 1994). By now meta-heuristics,
especially population-based ones such as GA’s, along with faster computers, can solve
complex optimization problems with unsmooth objective functions, even when simulation is
used to evaluate the objective function (Balamurugan, 2006; Haq and Kennan, 2006; Wang
and Schonfeld, 2005). In this paper a GA is used to find the optimal or near-optimal solution
to the selection and scheduling problem. To test this approach, a Frank-Wolfe traffic
assignment algorithm is used to compute the objective function. This algorithm can be
replaced later with a detailed simulation model.
A GA (Genetic Algorithm) is a metaheuristic method that imitates the biological
evolution and is based on the natural selection process (Michalewicz and Janikow, 1991). At
first, GAs create a set of possible solutions which form the “initial population”. This process
mostly creates the initial population randomly. A string of encoded genes called a
“chromosome” specifies each individual in the population. In this algorithm some individual
solutions with the best “fitness” value (i.e. objective function value) are chosen to reproduce
new offspring. This is usually a probabilistic process in which the individuals with better
fitness values have a higher probability of being selected for creating the next generation.
Then a series of mutation and crossover operators mate the selected solutions and change their
attributes to maintain the population’s diversity, and create the new generation (Golberg,
1989). In this study, each individual is defined as a string of numbers each corresponding to a
specific project to be implemented (FIGURE 2). In addition to random order solutions, a
greedy-order solution, a bottleneck-order solution form the initial population. In this context,
the greedy-order solutions represent the sequence of projects ordered by their benefit-cost
16
ratio, disregarding their interrelations. In bottleneck-order solutions, projects are ranked based
on their link volume-capacity ratios, which measure their congestion severity. This assumes
that more congested links should have higher priority for improvement.
The fitness function is equal to the value of the objective function which, as stated
earlier, is computed through the traffic assignment model. In maximization problems, the
selection probability corresponds to the value of the objective function. In minimization
problems the selection probability correlates inversely with the objective function value. To
avoid prematurity properties, a ranking method proposed by Michalewicz (1995) is used. In
this method the population is ordered from best to worst. Then, based the exponential ranking
value, the selection probability of each chromosome is assigned, assuming the lowest fitness
value is one (Michalewicz, 1995). Letting q be the selective pressure∈ [0,1], the selection
probability is defined as follows:
𝑃𝑖 = 𝑐 ∗ 𝑞(1 − 𝑞)𝑖−1, 𝑐 = 1/[1 − (1 − 𝑞)𝑃𝑜𝑝𝑆𝑖𝑧𝑒] (4)
Next, a roulette wheel approach is used to choose appropriate parents based on their
selection probabilities (Michalewicz, 1995). This process is conducted by spinning the
roulette wheel once for each individual in the population. Each time a random number r [0,1]
is generated, the 𝑖𝑡ℎ chromosome is selected so that 𝑤𝑖−1 < 𝑟 ≤ 𝑤𝑖 , where 𝑤𝑖 is the
cumulative probability for each chromosome. Then the crossover and mutation operators are
applied to reproduce offspring and create the new population. Common methods of mutation
and crossover are fairly inefficient for sequencing problems since they construct many
infeasible solutions with repetitive project numbers within one sequence. To avoid producing
such solutions, some other genetic operators are employed to solve the project sequencing
problem. These operators, adapted from Wang (2001), include Partial Mapped Crossover
(PMX), Position Based Crossover (PBX), Order Crossover (OX), Order Based Crossover
(OBX), Edge Recombination Crossover (ERX), Insertion Mutation (IM), Inversion Mutation
(VM) and Reciprocal Exchange Mutation (EM).
6. ANALYSIS FRAMEWORK The framework of the general proposed method for selecting, sequencing and scheduling
interrelated road projects is presented in Figure 1. The proposed combination of traffic
assignment and metaheuristic algorithms may be used to evaluate any sequence of projects
and find a near-optimal solution to the project selection and scheduling problem.
The pseudo algorithm provided in this section explains step-by-step how this problem is
tackled. First, the traffic assignment algorithm known as Frank-Wolfe, which is also used in
this study to evaluate the system at various stages, is described. This user equilibrium model
distributes flow in the network in a way that no individual user can reduce its trip cost by
switching routes. The second part describes the optimization algorithm. It also explains how
the user equilibrium algorithm is used within the GA to evaluate the objective function i.e.
fitness value of the population. In this case, each chromosome presents a string of numbers
which is the sequence of projects. The fitness value i.e. the objective function for each
chromosome is estimated by re-running the user equilibrium model at relatively short
intervals during the analysis period, and thereby estimate the effects of additional projects on
traffic volumes and speeds throughout the system.. This in fact captures the interrelation
17
among projects. Equation 1 yields the present value of total cost which is also the fitness
value for the chromosome. Accordingly, new generations are created and evaluated until the
GA’s termination condition is met.
Evaluation Model – User Equilibrium (Frank-Wolfe)
Given a current travel time for link a, 𝑡𝑎𝑛−1 the nth iteration of the convex combination
algorithm is summarized as follows:
1. Initialization: all or nothing assignment assuming 𝑡𝑎𝑛−1 which yields 𝑥𝑎
𝑛.
2. Updating travel time: use a BPR function 𝑡𝑎𝑛 = 𝑡𝑎(𝑥𝑎
𝑛) = 𝑡0(1 + 0.15 (𝑣
𝑐)4).
3. Direction finding:
- Find shortest paths using Dijkstra Algorithm based on 𝑡𝑎𝑛
- All or nothing assignment considering 𝑡𝑎𝑛 which yields auxiliary flow 𝑦𝑎
𝑛.
4. Line search: find 𝛼 that solves 𝑚𝑖𝑛 ∑ ∫ 𝑡𝑎(𝜔)𝑑𝜔𝑥𝑎
𝑛+𝛼(𝑦𝑎𝑛−𝑥𝑎
𝑛)
0𝑎 .
5. Move: set 𝑥𝑎𝑛+1 = 𝑥𝑎
𝑛 + 𝛼𝑛(𝑦𝑎𝑛- 𝑥𝑎
𝑛), ∀𝛼.
6. Convergence test: If a convergence criterion met, stop. Otherwise set n=n+1 and go to
step 1.
Optimization Model – Genetic Algorithm
1. t ← 0
2. Initial population: Set initial population [P(t)].
3. Evaluate population:
- For each chromosome (sequence of projects), run User Equilibrium after each
project (gene) is implemented.
- Obtain travel time 𝑤𝑖𝑗, volume, VMT.
- Compute the fitness value through eq.1.
4. While not termination, do
- Select parents [P𝑝(𝑡)]
- Reproduce offspring by crossover operators [P𝑐(𝑡)]← [P𝑝(𝑡)]
- Mutate [P𝑐(𝑡)]
- Create next generation [P(t+1)]
- t ← t+1
End.
5. Obtain optimized sequence of projects.
7. CASE STUDY In the literature, simple examples of related problems have been published, e.g. by Tao and
Schonfeld (2006). A more complex example, namely the Sioux Falls network (LeBlank et al.,
1975) is used as a case study here. Sioux Falls is the largest city in the U.S. state of South
Dakota. Its simplified network with 24 nodes and 76 links, shown in Figure 3, is used here for
testing purposes. It is assumed for this example that the demand grows exponentially over the
planning horizon:
𝑑𝑖𝑗𝑡 = 𝑑𝑖𝑗
0 ∗ (1 + 𝑟)𝑡 (5)
18
where 𝑑𝑖𝑗𝑡 is the demand between origin 𝑖 and destination 𝑗, 𝑑𝑖𝑗
0 is the base demand for the 𝑖𝑗
origin and destination (O/D) pair at time 0, and 𝑟 is the growth rate per period.
After running the traffic assignment model, the critical lanes with high volume-
capacity ratios are selected as an initial set of candidate projects. Our model allows volume-
capacity ratios above 1.0 since we use a BPR function for estimating link performances. Since
the demand matrix is symmetric for O/D pairs, each link expansion improvement is assumed
to be implemented in both directions between the two connected nodes, i.e. each project is
defined as expanding two links between a pair of connected nodes. This assumption is also
justified economically because it saves costs in using mobilized construction equipment and
other resources. To find appropriate initial solutions, the traffic assignment model is run for
all improvement scenarios. The first column in Table 1 shows the sequence of projects ranked
by their benefit-cost ratio in descending order. In this context, the benefit is the present value
of travel time savings, and the cost is the present value of implementation cost (greedy order
solution). The third column displays the sequence of projects based on their congestion
severity, where links with lower service levels have higher priorities (bottleneck order
solution).
TABLE 1 Greedy Order and Bottleneck Order Solutions
Greedy Order
Solution
(Link #)
Project Benefit
(dollar)
Bottleneck
Order Solution
(Link #)
V/C
Ratio
11 $217,300,346 11 2.17
36 $193,368,891 36 1.89
3 $189,404,178 34 1.79
12 $161,423,613 14 1.62
9 $117,425,401 9 1.59
15 $91,362,677 27 1.48
2 $87,751,583 35 1.42
25 $71,863,522 12 1.41
21 $70,811,860 15 1.36
4 $69,331,975 21 1.35
27 $68,775,533 3 1.35
37 $61,764,580 13 1.32
16 $61,099,054 30 1.31
22 $60,702,083 37 1.22
13 $60,135,953 22 1.21
14 $59,110,008 4 1.11
35 $44,182,898 2 1.11
30 $36,073,907 16 1.09
34 $5,242,573 25 1.04
After identifying an initial set of candidates, all projects are further investigated through a
benefit-cost analysis to identify and rank the initial economically beneficial projects. It is
assumed that each improvement project adds one lane, which is equivalent to 700
vehicles/hour additional capacity to each link, and the equivalent annual cost of each lane
19
expansion is assumed to be 4,000,000 $/lane-mile (Zhang et al., 2013). The main cost saving
of link expansion projects is the reduced travel time for all the users. These travel time
reductions can be computed through the traffic assignment model by comparing the total
system travel time before and after project implementation. Next, the previously described
GA is used to find near-optimal solutions for the sequence and schedule of selected projects.
When optimizing, we seek a sequence of projects which can be implemented within the
planning horizon (30 years). Therefore, every project with a scheduled completion time
beyond the planning horizon is eliminated from the sequence.
8. RESULTS As discussed previously, a traffic assignment model is used to evaluate the candidate projects
over the planning horizon and a GA is used to find near-optimal solutions. This section
analyzes the GA results and compares the basic scenario without improvement projects to the
scenarios with implemented projects.
TABLE 2 Optimal Sequence and Schedule
Optimal Sequence Completion Time
(year)
11 1.8
34 5.9
36 8.8
9 10.8
14 14.8
3 16.2
35 20.7
27 22.7
37 25.0
12 28.0
NPV of Total
Cost×106($) 8535.93
In this analysis the average GA running time per iteration is 300 sec and the entire analysis
takes about 8 hours to run.
Table 2 presents the optimal sequence and the corresponding schedule of projects
along with the objective value. The first column presents the link identifiers as ordered in the
optimized solution. As stated earlier, each link expansion improvement is assumed to be
implemented in both directions between the two connected nodes. Accordingly, the optimized
schedule is directly determined by the sequence of selected projects, assuming it is efficient to
fund and finish one project at a time, and gain its benefits as soon as it is completed. Thus, as
20
explained in section 2, successive projects in the sequence are completed when the available
cumulative budget equals the cumulative project cost. Figure 5 shows the accumulated total
delay costs for three scenarios: (i) no project implementation, (ii) project implementation
based on greedy solution, and (iii) optimized project schedule. These results indicate that at
the end of 30 years, the improvement projects can save up to 21% of the total delay costs
compared to no project implementation and 10.5% compared to the greedy order solution.
In addition to Sioux Falls network which is fairly small, this method is also applied to
the much larger Anaheim network, which is displayed in Figure 5. It has 416 nodes (of which
38 are origin/destination centroids), 914 links, and 1406 O-D pairs. All the network-related
information is extracted from (Bar-Gera, 2011). In this case, we tested the algorithm for 20,
40, 80 and 100 candidate projects. Table 3 compares CPU times for the Anaheim and Sioux
Falls networks. It can be seen that a larger network significantly increases the CPU time. The
results also indicate that the network size affects the CPU time much more than the number of
projects. In this case, where number of links in the Anaheim network is 12 times higher, the
CPU time per generation becomes almost 115 times higher. This occurs because the traffic
assignment algorithm has to evaluate the entire network regardless of the number of projects.
Also, the number of generations for comparable precision is likely to increase with network
size. In conclusion, this method is applicable to fairly large networks with numerous projects,
but computational improvements would be desirable for analyzing very large networks.
Table 3 CPU Time per Generation (Sec)
Sioux Falls Number of projects 5 10 15 20
CPU time 51.65 91.26 149.25 161.53
Anaheim
Number of projects 20 40 80 100
CPU time 10,472 12,764 16,897 18,533
9. ALGORITHM TESTING To evaluate the results emerging from this algorithm, an exhaustive enumeration is carried out
for the Sioux Falls network. Since the enumeration of the original problem with 20 candidate
projects (i.e. 20! possible solutions) is lengthy and requires extensive computation time, this
test is done for smaller problems with fewer projects. In this case, we consider four problems
with 4, 5, 6 or 7 projects to be ranked. Each case is solved both by the GA and by a complete
enumeration which evaluates each possible combination of projects and renders the exact
solution. The results presented in TABLE 4 indicate that the GA yields the exact solution
from enumeration in all four cases.
Table 4 Complete Enumeration Test
Complete enumeration GA solution
Number of
projects Solution space
Total system
cost * 106 Optimal
sequence Total system
cost * 106 Optimal
sequence
4 4!=24 90980 3,2,1,4 90980 3,2,1,4
5 5!=120 94248 3,2,5,4,1 94248 3,2,5,4,1
21
6 6!=720 98009 3,2,5,4,1,6 98009 3,2,5,4,1,6
7 7!=5040 99301 3,2,5,4,1,6,7 99301 3,2,5,4,1,6,7
In general, it is impractical to fully guarantee that the results of heuristic algorithms
are globally optimal, and it is somewhat difficult to assess the goodness of solutions obtained
by the evolutionary methods. In this study, a statistical experiment is conducted to examine
the effectiveness of the algorithm. For this purpose, first a sample of randomly generated
independent solutions is created. The next step is to fit an appropriate distribution to the
fitness values. The final step is to calculate the cumulative probability of the solution found
by the algorithm based on the fitted distribution. It is desirable to obtain a very low
probability to demonstrate the goodness of the solution. Accordingly, a random sample of
50,000 solutions is created, for which the objective function minimum is 8709.19×106 and
maximum is 15769.69×106. After exploring different distributions, the Lognormal (mu=
9660, sigma= 0.0248) distribution is found to yield the best fit. Figure 6 shows the fitted
distribution and the data derived from random sampling. It is evident that the minimum value
in the distribution of 50,000 random solutions is higher (costlier) than the optimal solution
presented in TABLE 2. In other words, the solution found by the algorithm excels all the
random solutions in the distribution.
The cumulative probability of the best solution found by the GA according to the
Lognormal distribution is 𝑝 = 𝐹(𝑥| μ, σ) = 𝐹(8535.93 × 106| 9660, 0.0248) = 3.597 ×10−5 which can be derived from the following equation:
` 𝑝 = 𝐹(𝑥| μ, σ) =1
𝜎√2𝜋∫
𝑒−(ln(𝑡)−𝜇)2
2𝜎2
𝑡
𝑥
0
𝑑𝑡 (6)
This result implies that the best solution obtained by the algorithm dominates 99.999% of the
random solutions in the distribution. Therefore, the solution found by the GA, although not
guaranteed to be globally optimal, is very good compared to other possible alternatives in the
solution space and the deviation from global optimality is likely to be very small compared to
uncertainties and errors in the problem’s inputs.
10. CONCLUSIONS The capacity expansion of links in road networks is a typical example of interrelated
alternatives for which the selection and sequencing of projects becomes a challenging
optimization problem with a “noisy” objective function surface. Common methods for
evaluating and prioritizing such problems are often incapable of capturing the interactions
among projects, and are mostly limited to pair-wise or at best n-wise interactions. The main
contribution of this study is to demonstrate how a traffic assignment model can be combined
effectively with a GA in a multi-period analysis for planning and prioritizing purposes while
capturing interactions among projects. We also design the algorithm to account for the
possibility that candidate projects may become economically justified or unjustified after the
implementation of previous projects. Another contribution is to reformulate the budget
constraint to include possible internal funding from fuel taxes. Also, we assume that the
demand changes during the planning horizon (growing exponentially in our example).
22
Finally, we demonstrate this methodology by conducting a case study and present a statistical
test of the goodness of the heuristic results.
In this study, a GA approach is employed here to optimize the selection and
scheduling of link expansion projects. The study uses a simple traffic assignment model to
evaluate the objective function and combines it with the GA to optimize the solution.
Although road expansion projects are the focus of this study, the proposed methodology
should be applicable to general cases involving more complex systems. More specifically,
GAs can optimize very intractable objective functions without requiring restrictive
assumptions about their structures. This allows analysts to effectively combine an appropriate
evaluation tool (e.g. microscopic simulation, simulation approximates, queuing or neural
networks, depending on the problem) with the GA, and to solve the planning and scheduling
problem for a variety of interrelated alternatives.
Future research may focus on developing general frameworks for solving the problem
of planning and prioritizing interrelated alternatives in a wide range of applications. Although
many components of such a general method exist, they could benefit from further
improvements. Accordingly, the work presented in this paper may be extended by
incorporating more complex evaluation models (e.g. micro simulation) to capture saturation
effects in networks. Future work may also account for uncertainties of important variables,
and consider other possibilities, such as multiple alternatives per location, facility changes
over time at the same location, and traffic delays during construction. Computational
improvements in the algorithm would be desirable, e.g. by distributing GA’s operators among
multiple computer processors. It may also be interesting to optimize particular projects
endogenously instead of selecting them from among pre-specified projects.
11. ACKNOWLEDGEMENTS
The authors are grateful for the comments provided by two reviewers. This work was partly
funded by the U.S. Department of Transportation through the National Transportation Center
at the University of Maryland.
23
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25
FIGURE 1 Framework of Optimization Process.
FIGURE 2 Example of a Feasible Solution.
26
FIGURE 3 Sioux Falls Network.
FIGURE 4 Accumulated Total Delay Cost with and without projects.
0
2E+09
4E+09
6E+09
8E+09
1E+10
1.2E+10
0.0 10.0 20.0 30.0
To
tal
Co
st (
$)
Year
without projects
Projects ranked bygreedy order solution
Projects ranked byoptimal solution
27
Figure 5 Anaheim Network.
FIGURE 6 Fitted Lognormal Distribution.
28
Appendix 2 - Jovanovic, U., Shayanfar, E. and Schonfeld, P. “Selecting and
Scheduling Link and Intersection Improvements in Urban Networks,”
Transportation Research Record, v2672, Dec. 2018, pages 1-11.
Selecting and Scheduling Link and Intersection Improvements in Urban Networks
Uros Jovanovic, Elham Shayanfar, and Paul M. Schonfeld
Department of Civil and Environmental Engineering, University of Maryland, College Park, MD
Corresponding Author:
Address correspondence to Paul M. Schonfeld: [email protected]
Abstract
Deciding which projects, alternatives or investments to implement is a complex and important
problem not only in transportation engineering, but in management, operations research and
economics. Projects are interrelated if their benefits or costs depend on which other projects are
implemented. Furthermore, in the network development problem analyzed here, the timing of projects
also affects the benefits and costs of other projects. This paper presents a method for optimizing the
selection and scheduling of interrelated improvements in road networks that explicitly considers
intersections. The Frank Wolfe algorithm, which is modified here to consider intersections, is used for
evaluating network improvements as well as for traffic assignment. Intersections are modelled with
pseudo-links whose delays are estimated with Akcelik’s generalized model. The objective is to
minimize the present value of total costs (including user time) by determining which projects should
be selected and when they should be completed. A genetic algorithm is used for optimizing the
sequence and schedule of projects.
29
For decades transportation engineers have been dealing with the problem of evaluating, selecting and
scheduling infrastructure projects. Considered alternatives can be classified as follows:
● Mutually exclusive: Only one alternative may be selected;
● Independent: The benefits and costs of alternatives
are independent of which alternatives are selected or when those are implemented;
● Interdependent (interrelated).
Interrelated alternatives pervade transportation net- works since improvements alter the flows, and
hence bene- fits, on other network components. This paper aims to show how a traffic assignment
model can be used to evaluate the objective function of an investment planning optimization problem
for an urban road network, especially by showing how intersections can be included in the traffic
assignment. A method is presented for evaluating, selecting and scheduling interdependent
improvement alternatives in urban road networks, which extends Shayanfar et al. (1) by considering
intersection improvements in addition to link widening alternatives. It is shown how a traffic
assignment model can be effectively modified to consider intersection flows and delays by introducing
pseudo-links. Adding pseudo-links for each of three movements (left, through and right) at each
approach of a four-leg intersection, creates a total of 12 pseudo-links per intersection. Moreover, a
traffic assignment model is shown to be effectively combined with a genetic algorithm for planning and
prioritizing purposes while considering interrelations among candidate projects. The background section
reviews some prior studies on intersection delay, selection and scheduling of project alternatives, and
traffic assignment. The next two sections present the evaluation model and the genetic algorithm used
for optimizing the project selection and schedule. A case study is presented on the Sioux Falls network
and the results obtained with the modified traffic assignment model and genetic algorithm in
optimizing the network development schedule. Conclusions and suggestions for extensions are
presented in the last section.
Background
Intersections are crucial components in urban road net- works since they affect traffic capacity and
delay at least as much as road links. Typically, four-leg intersections allow up to 12 legal vehicular
movements and 4 legal pedestrian crossing movements. Traffic signals assign right-of-way, and can
significantly reduce the number of conflicts, thus regulating the traffic flow. One of the many
disadvantages of traffic signals is the possibility of excessive delay which can congest the network,
which, in turn, increases cost, pollution and driver anxiety. Early studies on delays at signalized
intersections include Wardrop (2), who assumed that vehicles enter intersections with uniform arrivals,
and Webster (3) who studied delays for vehicles at pretimed signals and optimized their settings.
Delay relates to the amount of excess travel time, fuel consumption, and the frustration and discomfort
of drivers. Delay can also be used to compare the performance of an intersection under different
demand, control and operating conditions. For intersections, delay can be calculated simply, as the
difference in the departure time and the arrival time of a vehicle. Estimation of overflow delay is one
of the major difficulties in developing delay models at signalized intersections. The difficulty is
obtaining a simple and easily computable formula for overflow delay and has forced researchers and
analysts to search for approximations and boundary values. Numerous intersection delay models have
been developed, including Webster’s (3), Highway Capacity Manual (HCM) (4), Australian (variation
30
of the Akcelik delay model (5, 6), and Canadian (7). The delay model used here is Akcelik’s, because
it gives delay values close to the HCM formula for v/c \ 1.0, but with fewer assumptions about
parameters. It is expressed in Equations 1 and 2 as
where
d = average overall delay (sec/veh),
C = cycle time (sec),
l = fraction of the cycle which is effectively green for the phase under consideration,
x = v/c ratio,
T = flow period (h),
c = link capacity (veh/h),
m, n, a, b = calibration parameters, whose values are available for different delay models (e.g.,
Australian, Canadian, TRANSYT (8), and HCM) in Akcelik’s paper (5), and
s*g = capacity per cycle (veh/cycle).
Parameters n, m, a and b according to Akcelik’s papers (5, 6) have the following values
respectively: 0, 8, 0.5, and 0. Therefore, the two equations above become
The overall delay dI at an intersection can be calculated as
where dA is delay on approach A, and vA is volume on approach A. Heidemann (9) and Olszewski (10)
used probability distribution function to estimate delay at signalized intersections.In their models, the
probability distributions of delay were obtained from the probabilities of queue lengths.
31
Among many approaches used to tackle the problem of project selection and scheduling are integer
programming, used by Weingartner (11) and by Cochran et al. (12), and dynamic programming, used
by Weingartner (11) and by Nemhauser and Ulman (13). One notable study on interrelated projects is
Weingartner’s (11) which presents, among other problems, interdependent projects with budget
constraints.
Mehrez et al. (14) use a multi-attribute function to specify the decision maker’s preference with a
zero-one budget model to solve the problem of selection of interrelated multi-objective long-range
projects. The authors define a set of n indivisible projects contributing to m tangible and intangible
attributes with L limited resources available for T periods. In addition, they use a utility function with
m attributes and regard each project as a collection of subprojects, each one contributing to one of the
attributes affected by the projects.
Evaluation Model
Traffic assignment can be formulated as the problem of finding the equilibrium flow pattern over a
given transportation network, if its graph representation, the associated link performance function and
an origin–destination (O-D) matrix is known. Assignment of traffic flows on network links is a result
of equalizing transportation demand (O-D matrix) and transportation supply (link and node capacity,
management actions). A reasonable assumption is that all travelers try to minimize their own travel
time between their own origins and destinations. Other assumptions are that travel times increase with
link flows, and all individuals behave identically. User equilibrium (stable condition) is achieved when
no traveler can improve their travel time by changing route. Notable publications that dealt with traffic
assignment include Florian (15), Sheffi (16), and Boyce and Ran (17, 18). However, none of these
consider intersection characteristics and performance.
This paper applies the convex combination algorithm developed by Frank and Wolfe (19) to evaluate
link and intersection expansion projects upon their implementation in the network. The Frank-Wolfe
(FW) algorithm is an iterative algorithm used for solving a user equilibrium traffic assignment which
is a nonlinear programming problem with convex objective function and linear constraints. Given ta0 a
(initial travel time for link a), the convex combination algorithm is as follows:
32
Problem Statement
The problem considered here is NP hard (20) with a nonconvex objective function. The problem
grows rapidly as the number of candidate projects increases, and can be classified as a combinatorial
optimization problem. This type of problem involves finding values for discrete variables in such a
way that the optimal solution is found with respect to the objective function. Many practical problems
can be classified as combinatorial optimization problems such as the shortest path algorithm. Other
examples are the optimal assignment of employees to tasks to be performed and the traveling salesman
problem. Dorigo et al. (21) formulated a combinatorial optimization problem U as a triple (S, f, O),
where S is the set of candidate solutions (sequence of projects), f is the objective function (present
value of total costs) which assigns an objective function value f (s) to each candidate solution s 2 S,
and O is the set of constraints (budget constraint in our case). The solutions belonging to the set ~S S
of candidate solutions that satisfy the constraints O are called feasible solutions. The goal, according
to Dorigo et al. (21), is to find a globally optimal feasible solution s* (optimal sequence of projects).
In this study, the present value of total cost during the analysis period is the objective function, subject
to a budget constraint. The total cost consists of: (i) supplier cost, defined as the present value of all
project costs, and (ii) user cost, defined as the delay multiplied by the value of time. Accordingly, the
objective function can be formulated as
where
wij = waiting time on link i in year j,
ci = present value of the cost of link project i,
npl = number of link projects (link improvements),
nl = total number of links,
nI = total number of intersections,
npI = number of intersection improvement projects,
Ci = present value of the cost of intersection project i,
v = value of time, and
r = interest rate.
The cost of intersection project i can be written as
where
33
Cci =capital cost of improvement of intersection i ($/ft2),
Cpi = cost of pavement maintenance of intersection I ($/ft2),
AIi = area of the land needed to improve intersection I (ft2),
Ai = overall area of the intersection i (ft2)
The objective function is bound by the following cumulative budget constraint (22) as
where ti is the time when project i is finished, and xi(t) is a binary variable specifying whether project i
is finished by time t. Since in most realistic problems the cumulative budget constraint is binding, that
is there is never enough funding for all the available projects that are worth implementing, the
optimized project sequence represented by the set of all tis uniquely determines the schedule of
projects (1, 22).
Optimization Method
A genetic algorithm (GA) is a search technique inspired by biologic natural selection and evolution:
‘‘survival of the fittest’’. Traditional techniques evaluate only one potential solution at a time when
searching for the optimal solution, while a GA searches by concurrently examining a population of
solutions. First, the GA generates many different solutions and computes their fitness value (which in
most cases is the objective function value). Then, solutions are ranked based on their fitness value.
Solutions with better fitness values are saved, while others are discarded. Some saved solutions are
chosen as parents, and genetic operators, such as mutation and recombination operators, are applied on
them to create a new generation of solutions. This process is repeated, until the specified number of
generations is achieved or until the fitness function stops improving significantly. The GA includes the
following steps (23):
1. Code the problem and determine the values of the parameters.
2. Form an initial population which contains n strings, where n depends on the type of problem
examined. Evaluate the fitness function of every string.
3. Assuming the probability of choice is proportional to values of fitness function, choose n potential
parents.
4. Randomly choose two or more parents and apply operators such as recombination and mutation
operators to create offspring until a new population of n offspring is created.
5. Evaluate the fitness function for the new population for every offspring.
6. If the stopping criterion is reached, terminate the algorithm, and report the optimized solution (one
34
with the best fitness value). Otherwise, return to step 3.
In this study, the initial population of the GA is generated randomly and solutions are represented by
integer digits showing the sequence of the projects being implemented. Each individual in a
population is defined as a string of numbers, each corresponding to a specific project in a sequence.
The fitness function is the value of the objective function and is computed through the traffic
assignment model.
Figure 1. Graphical representation of the Sioux Falls network.
Case Study
The Sioux Falls network adopted from LeBlanc et al. (24) is used here as a case study. This network
differs from the real network since it mainly includes the city’s major arterials. It has been used in
many previous studies. Figure 1 depicts the Sioux Falls network with 24 nodes and 76 links.
35
After running the traffic assignment model on the Sioux Falls network, links and intersections (nodes)
that have critical volume-capacity (v-c) ratios are identified as an initial set of project improvements.
The BPR function (19) used as a link performance function allows v-c ratios to exceed 1.0, which
helps us identify the most congested links.
The project alternatives considered are link widenings (which are assumed to be applied
symmetrically in both directions between the two connecting nodes because the O-D table is
symmetric), and vertical, horizontal, or vertical and horizontal, improvements of intersections.
Improvements are carried through the entire intersection for consistency with the number of lanes on
the intersection’s legs; there are two types of improvements that are considered in this paper: (i) N-S
widening of the intersection between the North–South approaches, (ii) E-W widening of the
intersection between East–West approaches. It is assumed that some projects should be bundled
because it saves costs due to the joint use of resources and construction equipment. The assumption to
bundle some projects is justified economically because it saves costs due to the joint use of resources
and construction equipment.
In this example, it is assumed that the demand grows exponentially over the planning horizon as
where dt ij is the demand between origin i and destination j, d0 ij is the base demand for the ij origin
and destination (O-D) pair at time 0, and g is the growth rate per period. Some numerical values of the
input parameters and their units are displayed in Table 1.
Nodes 8, 11 and 16 represent two-phased intersections in the Sioux Falls network. Intersections were
modeled by adding one pseudo-link for each movement between link pairs, for example, for
intersection 8, link 47 there are three pseudo-links (47002, 47004, 47006) for three movements (left-
turn (002 part), through movement (004), and right-turn (006), respectively). Overall, for the three
intersections (8, 11 and 16), 36 pseudo-links are added to the network. Table 2 shows the pseudo-links
for intersections, their capacity, free flow travel time (t0), and which pseudo-link belongs to which
intersection. The capacity of each pseudo-link was set as the minimum value of the capacities of the
two real links it connects.
Table 3 shows the initial volumes for each of the O-D pairs. It is evident that there are no trips
originating and ending at nodes 8, 11, 16, because we consider them as intersections in the Sioux Falls
network. In Table 4, the values of delay on intersection pseudo-links, the volumes on each pseudo-
link, and pseudo v-c ratio are presented.
Figure 2 shows how the values of delay increase as the pseudo v-c ratio increases, for the three
pseudo-links 36004, 16002, and 52002. These pseudo-links were chosen because of their large
increases in delays as volume increases. The delay on each of the pseudo-links varies slightly, as can
be seen in Figure 2.
Figure 3 shows overall intersection delay for the three intersections as function of the percentage of
increase of the original O-D volumes, in 10% increments ranging from 10% to 140% of the original
O-D table. In it the intersection delay usually increases as the percentage of volume increases, with
intersections 8 and 16 having the greatest increases in delay. Due to traffic re-assignment, the delay
increase is not monotonic at individual intersections.
36
Intersections 8 and 16 are considered for improvement based on their delay values. The links to be
improved were chosen because of their high v-c ratios (above 0.6). Table 5 summarizes the list of
projects. Intersections with the highest delay values and links with the highest v-c ratio are selected for
improvement. Table 6 shows the bottleneck sequence and schedule of projects (ordered based on the
projects’ v-c ratios), greedy sequence and schedule (ordered based on their benefit-cost (b-c) ratio),
and the GA-optimized sequence and schedule of projects. In this case, benefit is defined as the
monetary value of total travel time savings from implementing one project, and cost is simply the
implementation cost of each project. The present values of total costs after each project
implementation are also shown in Table 6. These results indicate that the GA yields a better solution,
that is, with lower total cost compared to sequences based on b-c and v-c ratios. This occurs because
the GA process accounts for project interrelations, unlike common practices such as b-c ratio and
congestion level rankings.
Figure 4 shows the performance of the GA; the optimized solution is reached after 22 generations. The
stopping criterion for the GA was set at 10 successive similar solutions (shown in Figure 4) but, for
more confidence in the results, we let it run further for 200 generations, which yielded the same
solution. The CPU time for entire analysis is 3300 seconds. Table 7 demonstrates the sensitivity of the
optimized sequence, schedule, and the objective function value (total cost) to changes in demand.
Demand is changed by the same percentage for each cell in the O-D matrix. Table 7 also presents the
sensitivity of results to changes in the available budget. The variation in budget is specified as
different percentages of the original value, which was set to $1.5 million/year. It should be noted that
unsteady budget flows do not increase the model’s complexity or computation time.
37
38
39
Conclusion
The improvement of intersections and links in a network is just one example of interrelated
alternatives for which the selections and scheduling of projects becomes a challenging optimization
problem. This paper modifies the FW traffic assignment model to consider intersection flows and
delays. This is done by introducing pseudolinks to the network and applying Akcelik’s delay model.
The modified model is then incorporated within a GA loop to optimize the selection and scheduling
problem. Common prioritizing practices which are rankings based on b-c ratio and congestion level do
not produce the optimal sequence of projects because they disregard the interrelations among projects,
unlike the GA used here. This methodology can be applied more generally to other more complex
cases. GAs can optimize very intractable objective functions without requiring restrictive assumptions
about their structures which allows them to be efficiently combined with other evaluation tools, to
solve selecting and scheduling problems.
Future research may focus on extending the model by incorporating more detailed evaluation methods
(such as simulation models) to capture dynamic effects in congested networks that are missed by the
FW algorithm. Future model versions may also consider more elaborate intersection configurations,
control policies and cyclical variations in daily and weekly traffic. Gas could be solved considerably
faster by distributing the evaluation of population members among multiple processors. Moreover,
individual improvements (resurfacing, widening) could be grouped to form a project, bus traffic could
be traced along with passenger vehicles in the traffic assignment method, and different cost rates could
be assumed for different types of improvements implemented.
Acknowledgments
The authors are grateful for the support received from the National University Transportation Center at
the University of Maryland, which partly funded this work, and from TRB reviewers who provided
valuable suggestions and comments.
40
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Appendix 3 - Peng, Y., Li, Z. and Schonfeld P. “Optimal Development of Rail
Transit Networks over Multiple Periods,” accepted for Transp. Research Part A:
Policy and Practice, d Jan. 15, 2019.
Optimizing development plan of rail transit projects
over multiple time periods
Ya-Ting Penga,b
, Zhi-Chun Lia, Paul Schonfeld
b
aSchool of Management, Huazhong University of Science and Technology, Wuhan 430074, China
bDepartment of Civil and Environmental Engineering, University of Maryland, 1173 Glenn Martin Hall,
College Park, MD 20742, USA
Forthcoming in Transportation Research Part A: Policy and Practice
Abstract
This paper addresses the development of interrelated rail transit projects in urban rail transit networks
over multiple time periods. It extends the traditional network design problems by explicitly
considering the time horizon and interrelations among projects in rail transit networks. The proposed
model determines which projects in a rail transit network should be selected and completed at what
times (i.e., project selection, sequence and completion time), while jointly optimizing the evolving
headways of rail transit lines, in order to minimize the present value of the total cost. In addition to the
financial budget provided by relevant agencies (e.g., governments), we consider fare revenues
generated from the operations of previous completed projects as an internal source of funding for later
projects. A Genetic Algorithm (GA) is adapted to solve this model and tested on the transit network
development of Wuhan city in China. Sensitivity analysis is conducted to explore the effects on the
development plan of some important factors, such as travel demand and annual financial budget.
Findings are reported on the efficiency of the adapted GA approach as well as on the impacts of travel
demand and budgets.
Keywords: Rail transit network; development plan; correlated projects; financial budget constraint.
43
1. Introduction
The past decade has witnessed rapid growth in rail transit investments in China. According to the
latest report by the Chinese Urban Rail Transit Association (CURTA, 2017), by the end of 2017, 165
rail transit lines with a total length of 5033 kilometers were operating in 34 cities in mainland China.
Currently, 5636 km of rail transit lines are under construction, and 7305 km of rail lines were
approved but not yet built. These rail transit projects require huge investment costs. For example, the
capital cost of Wuhan Metro Line 2 was about RMB600 million per kilometer (RMB is the Chinese
currency “Renminbi”. US$1 approximates RMB6.51 as of January 1, 2018). However, the
government funds available for investment in rail transit projects are limited. The investment or
improvement of the rail transit lines is thus usually a multi-stage process.
As an example, Fig. 1 shows the gradual development process of the rail transit projects in Wuhan
(a city located in Central China) in the past dozen years. It can be seen that Wuhan’s rail transit
network gradually expands from one line in 2005 to seven lines in 2017. The corresponding total rail
line length grows from 34.57 km to 237 km. During the development process, the order and time of
the project implementations can significantly affect user cost and the investment efficiency in terms of
total cost. This raises an important question addressed here: how should we design an appropriate
development plan for rail transit projects within financial constraints over a planning horizon such that
the discounted total cost in the urban system is minimized?
In the literature, transportation infrastructure investment issues have attracted widespread interest
due to their practical importance. Table 1 summarizes some principal contributions to the related
problems, in terms of the type of infrastructure, consideration of time horizon, and consideration of
interrelations among projects. It can be seen from Table 1 that the existing studies mainly focused on
the general road network design problems with a discrete approach (see e.g., Wang et al., 2013; Zhang
et al., 2014; Wang et al., 2015), a continuous approach (see e.g., Li et al., 2012; Yin et al., 2014; Liu
and Wang, 2015), or in a hybrid way (see e.g., Luathep et al., 2011). These models usually aimed to
add new links or expand the capacities of the old ones in the network. Certainly, this is also an
important part of urban rail transit network development. However, the urban rail transit network
development problem is more complex than the general road network development problems due to
the design and operating characteristics of rail transit lines. In this regard, Gao et al. (2004) developed
a bi-level model to examine the interaction between the supply side and the demand side in a transit
network design problem. Farahani et al. (2013) provided a comprehensive review of urban
transportation network design problems.
However, most of these were static models focused on stationary states, which cannot address the
dynamic or progressive improvements of the rail transit system. It is well known that as the urban
economy and population grows, together with the development for the transit network, the demand for
the rail transit service may significantly increase. This increase can affect the rail services such as their
headways, operating costs and fare revenues. Hence, the development decisions for the rail transit
network should change, which in turn affect the system’s travel demand. Thus, the demand for rail
transit service, the operational condition and the network development decisions in one period are
significantly affected by the decisions made in the previous periods, and therefore, vary over the entire
time horizon. Consequently, it is important to incorporate the time dimension in the rail transit
network development problem such that interactions between the supply and demand over different
time periods can be taken into account.
44
So far, researchers have made considerable efforts to consider the time horizon in transport
network design problems. For example, Cheng and Schonfeld (2015) optimized the extension of single
rail line outward from a city center over time. Shayanfar et al. (2016) proposed an optimization
framework for selecting and scheduling interrelated projects in a road network. Sun et al. (2017)
explored the selection of public transit modes by costs and benefits analysis and considered essential
factors in a long-term planning process, such as economies of scale in rail extensions and future cost
discounting. More recently, Sun et al. (2018) extended the work of Cheng and Schonfeld (2015) by
developing a bi-level model to determine how many stations along a rail line should be completed in
different time periods, while considering demand elasticity. It should be noted that the previous
relevant studies only considered single rail line, expanded outward from a city center. No comparable
studies have been found for the more general rail transit network development problem.
In this paper, we extend the related studies to consider the gradual development process of urban
rail transit networks, while accounting for correlations among projects in the rail transit network over
different time periods. Here, a project means to invest in one segment or link in a rail transit network.
Correlations among projects occur when the benefits and costs of projects in the rail transit network
depend on whether and when other projects are completed. When a project is implemented, both the
user costs of the newly built segments and those of the completed segments change since the number
of OD pairs connected by rail lines and thus the demand for rail services increases. Growing travel
demand can decrease the train headways and thus the user costs of completed segments along the rail
lines. However, the operating costs increase due to the rail transit network expansion and decreasing
train headways. Consequently, the total cost change (or project benefit) due to project development is
not a simply linear summation of cost changes from individual segments, but a consideration of the
operating cost increases and the user cost savings from all segments in the network. The correlations
among projects significantly affect the investment decision and the development plan. Thus, it is
important to account for the correlations among projects in the transit network and their effects on the
system’s total cost.
In light of the above discussion, this paper proposes a model for optimizing transit network
development process over time by considering time-varying demand, financial constraints, and
interrelations among projects over time and space. There are two main contributions in this paper.
First, a novel model is proposed to determine the development process of rail projects in a rail transit
network with limited financial budget over a planning horizon. In the proposed model, the present
value of the total cost is minimized by optimizing the project selection, sequence and implementation
schedule. The effects of the newly completed projects on transit systems and the present value of the
total cost are explicitly explored by incorporating the correlations among projects over time and space.
In addition, the growth of the travel demand over time is effectively captured by a time-varying travel
demand function. The budget constraint includes possible internal funding, such as from the fare
revenue generated from the operation of the transit rail lines. In other words, in addition to externally
provided budgets, the fare revenue collected from the previous years is used as an internal source of
funding to finance the successive projects. Second, some important factors that affect the development
plan of the public transit projects and the present value of the total system cost are identified. Results
reveal that both the initial travel demand and annual financial budget can significantly affect the
development plan for a rail transit network. The proposed model can serve as a useful tool to guide the
development process of urban transit networks.
45
The remainder of this paper is organized as follows. The next section describes some basic
assumptions and the components of the models, including user cost and supplier cost. Section 3
presents the model for optimizing the development plan by determining which projects will be
selected, when these projects are completed, and the train headways in each period on the rail lines in
the network. A genetic algorithm (GA) for solving the proposed model is presented in Section 4. Next,
numerical examples are provided to illustrate the applications of the proposed model in Section 5.
Finally, Section 6 provides conclusions and recommendations for further studies.
2. Components of the model
2.1. Assumptions
To facilitate the presentation of essential ideas without loss of generality, some basic
assumptions are made as follows.
A1. The layouts of rail transit lines and station locations are assumed to be exogenously given, as
assumed in Cheng and Schonfeld (2015) and Sun et al. (2018). In fact, determining the layouts of rail
transit lines and station locations in an urban rail transit network is a major task of transit system
planning. In this paper, we focus on the future development plan for this pre-given transit network,
that is, determining which projects should be selected and when these projects should be invested over
a planning horizon.
A2. It is assumed that the sequenced projects can be invested once the financial budget is available.
We aim to explore the transit network development by considering financial feasibility over time.
Moreover, the system operations such as rail line length and train headways change if new projects are
completed. These assumptions have been adopted in various previous studies (see e.g., Wang and
Schonfeld, 2008; Shayanfar et al., 2016).
A3. Travel demand is assumed to be at a stationary state within each development period but varies
among periods. Here, period refers to the development state of a transit network. Specifically, when a
project is completed (i.e., the development state of the network changes), the current period ends and
the next period begins. Therefore, the duration of periods depends on the interval between the
completion of two successive projects, which is determined by the development plan and may vary
over different periods. It is assumed that travel demand in different periods increases due to
demographic trends, economic growth and network development. It is also assumed that the travel
demand between OD pairs which are already connected by rail lines increases at a higher rate than that
between unconnected OD pairs. In this paper, an exponential form of travel demand function is
adopted (as in e.g., Shayanfar et al., 2016; Cheng and Schonfeld, 2015; Sun et al., 2018).
A4. The present value of the total cost in the urban system is assumed to be the sum of the discounted
total cost over all development periods (see e.g., Shayanfar et al., 2016). In each period, the total cost
includes user cost and supplier cost. The supplier cost refers to the cost for providing transit service,
which includes the capital investment, network maintenance, and vehicle operating cost.
A5. It is assumed that until origin-destination pairs are connected by rail lines, their demands are
served by other modes (e.g. autos or buses), at a cost proportional to travel distance.
46
A6. In this model at most one rail route exists between any OD pair. In fact, except in central parts of
cities with very large rail networks, most rail trips have no alternative rail paths. This typical situation
can be seen in many cities, such as Atlanta.
2.2. User cost
Consider an urban rail transit network ( , )G N A , where N is the set of nodes (transit stations or stops)
and A is the set of transit line segments in the network. Let W be the set of origin-destination (OD)
pairs in the network, L be the set of transit lines and T be the set of development periods. The binary
decision variable can be defined as
( )1, if segment already exists in period , , ,
0, otherwise.
t
a
a t a A t Ty
(1)
It should be noted that in the rail transit network, segment a may include several stations. This is
consistent with actual practice because it can yield economies of scale and save costs in using
mobilized resources such as construction equipment.
Let ( )
1
t
ac and ( )
2
t
ac be the user cost on segment a by rail and by other modes in period t, respectively.
The travel cost by rail consists of waiting cost and in-vehicle time cost. Note that the access cost that
be omitted because we assume that the station locations are predetermined (see Assumption 1). Thus,
we have
( )( )
1 1 2 , , , ,2
tt a al l
a
d hc a A l L t T
V
(2)
where 1 and 2 are the values of in-vehicle time and waiting time, respectively. al is a 0-1
indicator, which equals 1 when segment a is a section of rail line l, and 0 otherwise. ad is the length
of segment a, V is the average speed of trains, and ( )t
lh is the average train headway of rail line l
where segment a is located in period t. According to Assumption 5, the user cost on segment a by
other modes, ( )
2
t
ac , can be expressed as
( )
2 0 , , ,t
a ac c d a A t T (3)
where 0c is the cost per km of travelling by other modes, which is assumed to be a constant. It can be
seen from Assumption 5 that, before segment a is implemented or connected to rail lines, persons
passing through it have to choose other travel modes. Let ( )t
ac be the user cost on segment a, which
can be expressed as
( ) ( ) ( ) ( ) ( )
1 2= 1 , , ,t t t t t
a a a a ac y c y c a A t T (4)
where ( )t
ay is the decision variable, defined in Eq. (1), indicating whether segment a is
completed in period t.
47
The daily traffic volume on segment a in period t, ( )t
aQ , can be expressed as
( ) ( ) ( ) , , , ,t t t
a w wa
w W
Q q a A w W t T
(5)
where ( )t
wq is the daily travel demand between OD pair w in period t. ( )t
wa is an indicator, which equals
1 when segment a is on the route between OD pair w in period t, and 0 otherwise. Note that there is at
most one rail route connecting OD pair w (see Assumption 6). Therefore, the route index is omitted
here. We assume the travel demand increases over time due to demographic and economic growth and
network development. According to Assumption 3, the exponential form of travel demand function
can be expressed as
( ) (0)
1 21 (1 ) , , ,t t wx x xt
w w wq q g g t T w W
(6)
where (0)
wq is the daily travel demand between OD pair w in period 0, 1g is the base growth rate per
year due to demographic and economic growth and 2g is the additional annual growth rate when OD
pair w is connected (see Assumption 3). w is a 0-1 indicator, which equals 1 when OD pair w is
connected, and 0 otherwise. tx is the starting time of period t, and wx is the first time to complete the
connection for OD pair w. Let ( )t
uC be the annual user travel cost in period t. Thus, we obtain
( ) ( ) ( ) , , ,t t t
u a a
a A
C Q c a A t T
(7)
where is the average number of days of travel per traveler per year, which is used to
transform the daily basis cost to the yearly one. ( )t
aQ is the daily traffic volume on segment a
in period t and ( )t
ac is the user cost on segment a.
2.3. Supplier cost
According to Assumption 4, the cost of providing the rail transit service in each period includes the
capital investment cost of the new project, the maintenance cost of existing rail lines in this period,
and the vehicle operating cost in this period. Let ( )t
c be the capital investment cost in period t, ( )t
m
be the annual maintenance cost in period t, and ( )t
o be the annual vehicle operating cost in period t.
The capital investment cost ( )t
c in period t, such as land acquisition, design, and construction costs,
can be expressed as
( ) ( 1) ( ) , ,t t t
c a a a
a A
y y t T
(8)
where a is the capital investment cost for segment a. Note that the capital investment cost only
occurs at the time when segment a is developed (i.e., the end of this period and the beginning of the
48
next period). Here, the term ( 1) ( )t t
a ay y indicates whether or not segment a is selected at the end of
period t. It equals 1 when segment a is implemented in period t, and 0 otherwise.
The maintenance cost ( )t
m per year in period t, is directly proportional to the total length of the
existing transit lines in period t, which can be expressed as
( ) ( ) ,t t
m a a
a A
y d
(9)
where is maintenance cost of transit lines per kilometer per year.
The annual vehicle operating cost is the sum of the vehicle operating cost of each transit line.
Specifically, the annual vehicle operating cost of a transit line is its fleet size multiplied by annual
operating cost per train. To obtain the fleet size, the transit round trip time should be derived first. Let ( )t
lR be the round trip time of line l in period t and ( )t
lF be the fleet size of transit line l in period t.
Thus,
( ) ( )2 , , ,t t
l a a al
a A
R d y V l L t T
(10)
( ) ( ) ( ) , , ,t t t
l l lF R h l L t T (11)
where al is a 0-1 indicator determining whether or not segment a is a section of rail line l, defined in
Eq. (2). ( )t
a a al
a A
d y
is the length of line l completed in period t, which may change due to the
network development. Let be the operating cost per train per year. Therefore, the total yearly
vehicle operating cost of the system in period t ( )t
o can be expressed as
( ) ( ) , .t t
o l
l L
F t T
(12)
The rail line’s headway varies with its travel demand. Consequently, the headways are steady in each
development period, but vary among periods, like the changes in travel demand (see Assumption 3).
Therefore, we have to re-optimize the headways in each period, i.e. after every decision made. The
optimal headway for rail line l in period t ( )t
lh , can be determined by minimizing the total cost of the
system in this period. Specifically, the system’s total cost in period t is defined as the sum of the user
cost and the supplier cost in this period. Let ( )t be the total cost of the system in period t. According
to Eqs. (4)-(12), it can be expressed as
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1) ( )
1 21 , ,t t t t t t t t t t t
t a a a a a a a l l a a a
a A a A l L a A
Q y c y c y d R h y y t T
(13)
1 , ,t t tx x t T (14)
where t is the duration of period t, which is determined by the difference between the start time +1tx
of period t+1 and the start time tx of period t. In square brackets in the right hand side of Eq. (13) the
49
first term represents the annual user cost in period t, the second term is the annual maintenance cost in
period t, and the third term is the annual operating cost in period t. Setting
( )
( )0
t
t
lh
, we can
analytically obtain the optimal headway of transit line l in period t as
( )( )
( ) ( )
2
2, , ,
tt l
l t t
a a al
a A
Rh l L t T
y Q
(15)
where is the operating cost per train per year and 2 is the value of waiting time. ( ) ( )t t
a a al
a A
y Q
is traffic volume of line l in period t. Eq. (15) implies that the optimal headway of transit line l in
period t, ( )t
lh , decreases to accommodate the increased demand of this line over time.
3. Model formulation
As previously stated, the goal is to minimize the present value of the total cost by determining which
projects should be developed and when these projects should be completed. The discounted total cost
is the sum of the discounted total cost in each period. According to Eqs. (4)-(13), the model can be
formulated as follows.
( )
( ) ( ) ( ) ( ) ( ) ( 1) ( )
( )
,
min ,1+ 1+
t t ta t
t t t t t t t
t a a a a l l a a ata A a A l L a A
x xy x t T t T
Q c y d R h y y
r r
(16)
s.t.
( ) ( ) ( 1) , , , , =0,1,2,..., 1,t t t
i j az z y i j N a A t T (17)
( ) ( ) , , , ,t t
i az y t T a A i N (18)
( ) ( ) , , , ,t t
j az y t T a A j N (19)
( 1) ( ) , , =0,1,2,..., 1,t t
n nz z n N t T (20)
( 1) ( ) , , =0,1,2,..., 1,t t
a ay y a A t T (21)
( ) ( )
( ), , , ,t t veh
a a al ta A l
Ky Q a A t T l L
h
(22)
( ) ( ) ( ) , 0,1,2,..., ,t t t
cB t T (23)
( )
0 1= , 1,2,3,..., ,t
t tB B x x t T (24)
50
where r is the discount rate. The denominator (1+r) in Eq. (16) is used to convert the cost of future
investment to today’s cost. ( )t
ay and tx are the decision variables defined in Eqs. (1) and (6),
respectively. Eq. (16) is the objective function that minimizes the present value of the system’s total
cost. ( ) ( ) , 1,2,...,t t
nz n N z is the vector of 0-1 variables indicating whether a node is completed
in period t. i and j denote the indices for the two end nodes of segment a. Constraint (17) expresses the
segment connectivity in the network, which implies that the segments to be built should have at least
one end node already completed (i.e., the newly built segments must connect to the segments that have
been already completed). This constraint ensures that the network’s rail lines are extended by
connecting to the existing lines. However, there is an exception. Initially, when none of nodes or
segments in the network are yet completed, i.e., no existing lines need to be connected, any projects
may be considered for immediate implementation without subject to Constraint (17). Constraints (18)-
(19) mean that if segment a is completed, its two end nodes are also completed. Constraints (20) and
(21) are realistic constraints ensuring that after nodes and segments are completed, they always remain
in service in later periods. is the peak-hour factor, i.e., the ratio of peak-hour demand to the daily
demand, which is used to convert the passenger volume from a daily basis to an hourly basis. vehK is
the capacity of vehicles (i.e., the maximum number of passengers allowed in a vehicle, both seated and
standing). Constraint (22) is the line capacity constraint, which guarantees that the rail service supply
satisfies the associated (peak-hour) passenger demand. ( )tB is the budget flow in period t and 0B is
the annual budget level provided by relevant agencies (e.g., governments). Constraint (23) is a
reformulated budget constraint which considers an internal funding source, such as the rail fare
revenue collected from the rail service operations. The left-hand side of Constraint (23) denotes the
total available funding at the end of period t and the right-hand side denotes the capital investment cost
needed. The reformulated budget constraint reflects interrelations among projects in the transit
network since the capital used for development is partly supplied by fare revenue collected from the
rail operations, which may change with the network development. ( )t denotes the fare revenue
collected from the rail operations in period t, which can be expressed as
( ) ( ) ( )t t t
t a a a
a A
y Q fd
， (25)
where t is the duration time of period t, defined in Eq. (14). The fare on segment a is the fare per km
f multiplied by its length ad .
It should be noted that if the budget is limited throughout the planning horizon, i.e., never sufficient
for all beneficial projects, a project sequence uniquely determines a project schedule. The available
funds should always be used whenever they suffice to complete a project (see Assumption 2). Hence,
after the sequence of projects is determined, the completion time of these projects can be obtained by
checking budget constraint. Accordingly, only those projects whose implementation times are within
the planning horizon are selected. Here, the projects that are completed at the time beyond the
planning horizon are implicitly rejected. Thus, the development plan is optimized by first optimizing
the sequence of projects, and then determining the completion time of each project.
51
4. Solution algorithm
The above total cost minimization model (16)-(24) is a constrained integer programming problem,
which is non-linear and non-convex, making it difficult to find its globally optimal solution. A GA
approach is presented in this section due to its suitability for very “noisy” objective functions. GA’s
are inspired by phenomena in evolutionary biology. In a GA, a solution of the problem is called an
individual. It is represented as a sequence of variables called a chromosome or gene string. A group
including multiple individuals is defined as population. The essence of GA is population evolution
through selection, crossover and mutation. Generally, a GA starts from initializing a set of individuals,
i.e., a population, and then selecting the better individuals to reproduce offspring by applying genetic
operators such as crossover and mutation operators. As a result, the most adapted individuals survive
and thus the population can converge toward an optimized solution.
The GA in this paper is developed from basic GAs but differs from them in many ways. First, an
efficient genetic encoding scheme is adopted to deal with the constraints. Since the proposed model
has the network connectivity constraint (see Eq. (17)), traditional representation schemes such as the
sequence of projects may generate too many infeasible solutions. A general remedy for this problem is
to add penalty terms to fitness functions or use repair operators to transform infeasible solutions into
feasible ones. However, these methods cannot handle the connectivity constraint efficiently and
degrade the search efficiency in terms of speed and accuracy. Therefore, a novel genetic encoding
scheme is needed. Second, solutions capturing the characteristics of the network and projects are
incorporated into the initial population to accelerate the convergence of the GA. For example,
solutions that represent the sequence of projects ordered by their demand level and investment cost are
included in the initial population. Intuitively, development of projects with higher travel demand and
lower investment cost can contribute more to the system cost saving and thus those projects have
higher priority for development. As a result, such solutions may make better use of existing
information, which help accelerate the convergence of the GA. Third, some mechanisms are designed
to avoid GA prematurity. In the selection process, a ranking method is used to help the GA escape
from local optima. In addition, the catastrophe mechanism is introduced when the optima remain
unchanged for a certain number of generations (e.g., 50 generations). These mechanisms are capable
of enhancing the accuracy and stability of the GA.
4.1. Genetic encoding and decoding
The process of encoding a chromosome into a string is called genetic encoding and the process of
decoding a chromosome into a feasible solution to the problem is called decoding. In this paper, each
individual has one chromosome, which is encoded by a string of numbers representing the selection
priority of a specific project to be completed. Let 1 2, ,..., JE e e e be a chromosome represented by
a string of genes, where J is the number of possible projects to be selected. , 1,2,..., ,ie i J is the ith
gene on chromosome E, and its value indicates the selection priority of the ith project. The selection
priority for each project is randomly generated within 1, J exclusively. Thus, to initialize a
chromosome (i.e., an individual) is to generate J random numbers within 1, J . An example of a
chromosome is shown in Fig. 2.
52
The main idea of decoding is to choose the one with the highest selection priority value from the
candidate set as the successive project to be implemented. In this paper, a connectivity information
matrix Mark[i][n] is constructed to store whether node n is at the end of segment i (i.e., project i),
where 1,2,..., ,i J and 1,2,..., .n N N is the number of nodes in the transit network. Besides, a
vector I is used to indicate whether a node is completed. A procedure to generate a feasible solution to
the problem from a chromosome is displayed as follows.
Step 1. Initialize the candidate set by including all the feasible projects.
Step 2. Choose the project with the highest selection priority value from candidate set.
Step 3. Update vector I by checking constraints (18)-(19).
Step 4. Update the candidate set by deleting the projects that have been already completed and making
changes by checking Mark and Constraint (17).
Step 5. Check whether the candidate set is empty. If so, stop and output the sequence of projects to be
completed. If not, repeat steps 2-4.
It should be noted that since the values of selection priority for projects are distinct, each chromosome
can uniquely determine a feasible sequence of projects. As discussed in the last paragraph in Section 3,
a feasible sequence of projects can eventually determine a development plan. Therefore, each
chromosome can be uniquely decoded into a feasible solution to the problem. With this genetic
encoding scheme, all feasible solutions can be represented by changing the sequence of project
priorities.
To further illustrate the process of decoding, we consider a transit network in Fig. 3 and decode the
chromosome in Fig. 2 into a feasible solution to this network development problem. At the beginning,
initialize the candidate set as (1, 2, 3, 4, 5, 6). Then, choose project 1 from the candidate set as the first
project to be implemented due to its highest selection priority, so that the nodes (1, 3) are completed.
According to Constraint (17), only projects that connect to segments that have been already completed
can be included in the candidate set. Thus, we update the candidate set as (2, 3, 4). Choose project 4 as
the successive project because we have 4 (the selection priority of project 4)>3(the selection priority
of project 2)>2(the selection priority of project 3). Repeat those steps until the candidate set becomes
empty, so that we can obtain a unique feasible sequence of projects as (1, 4, 6, 2, 3, 5).
4.2. Calculating the fitness value
Before calculating the fitness value of an individual, we have to translate a chromosome (e.g., E= (6,
3, 2, 4, 1, 5) in Fig. 2) into a feasible sequence of projects (e.g., (1, 4, 6, 2, 3, 5)). In this paper, the
fitness function is equal to the value of the objective function as shown in Eq. (16). Therefore, the
fitness value of an individual is the discounted total cost of a project sequence. Let be the planning
horizon. The steps are displayed as follows.
Step 0. Initialization. Let t be the counter of periods and set t = 0.
53
Step 1. Calculate the travel demand for OD pairs in period t ( )t
q by Eq. (6). Then, determine the daily
traffic volume on segments ( )t
Q by Eq. (5), headway of transit lines ( )t
h by Eq. (15) and Constraint
(22), annual user cost ( )t
uC by Eq. (7) and annual supplier cost by Eqs. (8)-(12), respectively.
Step 2. Calculate the implementation time of the next project ( 1)tx
by checking budget constraint in
Eq. (23). If ( 1)tx , let
( 1)tx .
Step 3. Obtain the duration time of period t t by Eq. (14). Then, calculate the discounted total cost in
period t ( )t by Eq. (13) and the cumulative discounted total cost by Eq. (16).
Step 4. If ( 1)tx holds, set t=t+1 and go to step 1. Otherwise, stop.
4.3. Selection
Parents are chosen from the population according to a probability which correlates inversely with the
fitness value of individuals. To avoid prematurity of the GA, a ranking method proposed by
Michalewicz (1996) is adopted. In this method, we first order the individuals in the population from
best to worst according to their fitness values, i.e., the individual with the lowest fitness value is the
best and is ranked first. Then, we calculate the selection probability of each individual based the
exponential ranking value by assuming the lowest fitness value is one. Let 0p be the selective
pressure, which is a positive value between 0 and 1, i.e., 0 (0,1)p , and ip be the selection
probability of the individual ranked at i. Then, ip can be expressed as
1
0 0 0(1 ) 1 (1 ) ,i M
ip p p p (26)
where M is the population size. Next, a roulette wheel approach is used to choose appropriate parents
based on their selection probabilities. This process is conducted by spinning the roulette wheel once
for each individual in the population. Each time a random number (0,1)b is generated, the i_th
individual will be selected if 1i ib , where i is the cumulative probability for each individual.
4.4. Operators
It should be noted that common methods of mutation and crossover are fairly inefficient for our
problem since they construct many infeasible solutions with repetitive numbers within one
chromosome. To avoid producing such solutions and improve the efficiency, we adopt Partial
Matched Crossover (PMX) as the crossover operator and Reciprocal Exchange Mutation (REM) as the
mutation operator. These operators are explained by Wang (2001), and thus omitted here.
In general, GA has a strong local search ability, but may get trapped in local optima, which is also
known as prematurity. Therefore, the catastrophe mechanism is introduced (Gu et al., 2009). The main
idea of this mechanism is to discard the current optima so that the population may produce better
54
solution. The specific approach in this paper is to regenerate the initial population randomly when the
optima stay unchanged over a specified number of generations.
5. Numerical study
In this section, numerical examples are used to illustrate the applications of the proposed model and
the contributions of this paper. We consider the urban rail transit network represented in Fig. 3
composed of 3 transit rail lines, 7 nodes (represented by circles) and 6 segments between them. To
complete the development of this network, 6 candidate projects are considered. Specifically, each
project includes the development of one segment and the two end nodes of this segment (if they are
not yet completed). The input data for segments such as length, investment costs and associated rail
line are displayed in Table 2. Table 3 shows the daily travel demand between OD pairs. In the
following analyses, unless specifically stated otherwise, the input parameters and their baseline values
used in the model are the same as those shown in Table 4. We set the planning horizon as 10 years, the
annual capital budget as $250 million and the genetic parameters as follows: population size, pop_size
= 10; maximum generation, max_gen = 100; crossover probability, 0.8cP ; mutation probability,
0.5mP ; the number of implementing catastrophe mechanism, 1cn . The proposed solution
algorithm is coded in MATLAB and run on a ThinkPad Carbon X1 computer with an Intel(R)
Core(TM) i5 CPU (2.4 GHz) and 8 GB of RAM. This numerical experiment takes about 0.8 seconds
of CPU time.
5.1 Example 1
5.1.1 Optimized solution for rail transit development plan
Table 5 displays the optimized development plan of rail projects and the system performance. It can be
seen in Table 5 that 3 projects are selected over a planning horizon of 10 years, i.e., projects 4, 6, and
3, and they are completed sequentially at years 6.00, 8.61 and 9.47, respectively. Over time, the
headways of rail lines decrease, but the demand for rail service and discounted cumulative total cost
saving increase. Specifically, the headway of Line 1 decreases by 0.13 min from 1.54 min in period 1
to 1.41 min in period 3, and the headway of Line 3 decreases by 0.05 min from 2.91 min in period 2 to
2.86 min in period 3. However, the daily demand for rail service increases by 624.2 thousand from
585.60 thousand riders in period 1 to 1209.80 thousand riders in period 3, and the discounted
cumulative total cost saving increases by $8.04 billion from $7.21 billion to $15.25 billion. This
occurs because the development of the rail transit network increases the connectivity of OD pairs and
hence the demand for rail service, thereby decreasing headways (see Eq. (15)). Thus, the user costs
and total costs are reduced and the total cost saving increases.
Fig. 4 shows the changes of the state of the rail transit network over time with the development
plan. The bold segments represent those which are already in service in a period. It should be noted
that the initial state (from year 0 to 6.00) in which no segments are completed is displayed in Fig. 3.
Fig. 4a shows the state of the network in the first period, i.e., from year 6.00 to 8.61. In this period,
segment 4 is completed and in service. In period 2 from year 8.61 to 9.47, segment 6 is implemented
and connected to segment 4. Both segments 4 and 6 provide rail services, as shown in Fig. 4b. Fig. 4c
55
indicates that segment 3 is completed at the beginning of the third period and in service from year 9.47
to 10. It can be seen from Fig. 4 that throughout the planning horizon, the rail transit network
progressively expands to 3 rail lines with a total length of 41 km (i.e., sum of the length of segments 4,
6 and 3).
Fig. 5 shows the changes of discounted cumulative total cost with and without the rail transit
investment. It can be seen in Fig. 5 that the total cost curve with investment is under that without
investment after year 6.00. This means that the rail transit investment efficiently decreases the total
cost of system. It should be noted that in year 6.00, the discounted cumulative total cost with
investment is slightly above that without investment due to the capital investment cost of segment 4.
Fig. 5 also shows that over the planning horizon, the network development decreases the total cost
from $117.53 billion to $102.28 billion.
In order to verify the solution obtained by the proposed GA, we conduct a complete enumeration
for the urban transit network shown in Fig. 3. The comparsions of the results are displayed in Table 6.
Clearly, the solution obtained by the GA in this paper is consistent with that obtained by complete
enumeration. In addition, to test the convergence and stability of the proposed GA, the program is run
by 10 times. The results show that each run of the program converges to the same solution. This
demonstrates that the proposed GA has good stability. Therefore, we can conclude that the proposed
GA is capable of finding a very good and stable solution at acceptable computation cost (i.e., 0.8
seconds vs. 15 seconds).
5.1.2 Sensitivity analysis
To explore the effects of the initial travel demand on the optimized development plan and system
performance, we conduct numerical experiments by scaling the basic value of (0)
wq in Eq. (5) by 0.5
down and 1.5 up. Table 7 shows that as the travel demand increases, the number of implemented
projects and the total cost saving increases. Specifically, as the initial travel demand increases from (0)0.5 wq to
(0)1.5 wq , the number of projects selected increases from 2 to 4 and the total cost saving
increases from $6.22 billion to $28.62 billion. This is because higher fare revenue can be collected
from the operation of completed projects with higher demand, which increases the available budget for
network development. Thus, both the number of implemented projects and the total cost saving
increase.
Table 8 shows the changes of the optimized development plan with the annual budget level 0B in
Eq. (24). It can be noted in Table 8 that the annual budget level has a significant effect on the
optimized development plan and system performance in terms of the number of projects selected, the
time of implementation and the total cost saving. Specifically, as the annual budget level increases
from 00.8 B to 01.2 B , the number of projects selected increases by 3 from 1 to 4, the first
investment time decreases by 4 years from year 9.00 to year 5.00 and the total cost saving increases by
$22 billion from $0.95 billion to $22.95 billion. This implies that a higher budget level can accelerate
the development process and save more costs.
56
5.2 Example 2
To further illustrate the applications of the proposed model and test the performance of the GA on a
more complex problem, we apply the proposed model to the rail transit network development of
Wuhan city in China. As shown in Fig. 6a, there are 3 rail lines represented by three colors: blue for
Line 1, purple for Line 2 and green for Line 4. A rail transit network with 14 nodes (represented by
circles) and 13 segments between them is considered, as shown in Fig. 6b. Similarly, we consider the
development of one segment and its two end nodes (if they are not yet completed) as a candidate
project. The input data for segments and OD pairs are displayed in Tables 9 and 10, respectively. The
base values of the input parameter are shown in Table 4. We set the planning horizon as 15 years and
the annual budget flow as $1 billion. The genetic parameters are: population size, pop_size = 50;
maximum generation, max_gen = 500; crossover probability, 0.8cP ; mutation probability,
0.2mP ; the number of implementing catastrophe mechanism, 20cn ; and run 10 times. This
numerical experiment requires an average CPU time of about 13 min. Using the proposed GA, we can
obtain the same solution for all runs, which shows that the proposed GA maintains its stability on a
more complex problem.
The optimized development plan and headways of rail lines are displayed in Table 11. It can be
seen that 11 projects are developed over a planning horizon of 15 years with a total cost of $99.28
billion. Specifically, projects 3, 6, 8, 10, 4, 2, 5, 9, 11, 7 and 1 are completed in sequence at years 0.29,
1.60, 3.63, 4.67, 6.45, 7.98, 10.17, 11.40, 12.06, 12.76, 13.86, respectively. This result is roughly
consistent with the realistic development of the urban rail transit network in Wuhan between 2000 and
2014, as shown in Fig. 1.
Since the enumeration of this problem with 13 candidate projects (i.e. 13! possible solutions)
requires extensive computation time, and no existing method can guarantee a globally optimal
solution, it is difficult to verify the solution obtained by the proposed GA. In this paper, a statistical
method is adopted to evaluate the solution (as in Jong and Schonfeld, 2003 or Shayanfar et al., 2016).
The main steps are as follows. First, a large sample of solutions is randomly generated. These
solutions should be representative and independent of each other to ensure the generality of the
sample. Then, the fitness values of the solutions in the sample are calculated. Next, a distribution is
fitted to the fitness values and checked with Chi-Square or K-S tests. It should be noted that the fitted
distribution should approximate the actual distribution of fitness values for all possible solutions in the
search space due to the representativeness and randomness of the sample. Finally, the cumulative
probability of the solution in the distribution can be calculated. This cumulative probability represents
the probability that is the other solutions in the distribution smaller than the obtained solution.
Therefore, the lower the probability, the better the solution.
In this paper, a sample size of 100,000 independent solutions is randomly generated, for which the
minimum of the fitness values is 99.88×109 and the maximum is 131.21×10
9. Note that the best
solution found by the proposed GA is 99.28×109 which is better than any of the 100,000 randomly
generated solutions, as shown in Fig. 7. The distribution of the fitness values for the solutions in the
sample is supposed to cover the fitness values for all possible solutions in the search space. Actually, it
does not. This means that better solutions (i.e., having lower fitness values) are extremely rare for this
example and are unlikely to be included in a randomly generated sample. The best fitting distribution
among those searched is the generalized extreme value distribution, i.e., GEV(9112.5 114 0 ,
57
5.27436 , 0.145511 ), as is shown in Fig. 7. Its probability density function can be
expressed as
1 ( )1( ) ( ) xf x x e
, where
1
( )
1 , if 0,( )
, if 0.x
x
x
e
(27)
The cumulative probability of the best solution found by the proposed GA (i.e., 99.28×109 in Table
11) can be calculated by integrating ( )f x from 0 to 99.28×109. The result is 2.0552×10
-4, which
means that the solution obtained by the proposed GA dominates 99.98% of the solutions in the
distribution, as well as 100% of the 100,000 randomly generated solutions. That is to say, the best
solution found, although not guaranteed to be globally optimal, is still remarkably good when
compared with other possible solutions in the search space. This suggests that the accuracy of the
proposed development scheduling method is limited far more by the accuracy of input data than by the
optimization capability of the GA.
6. Conclusions and further studies
To address the dynamic development problem of urban rail transit networks with limited budgets, this
paper proposes a novel model to optimize the development plan of rail transit projects over a planning
horizon. The proposed model determines which projects should be implemented and when to complete
these projects together with train headways by minimizing the present value of the total cost. The
time-varying demand and the interrelation among projects are explicitly considered. Specifically, the
model captures how the travel demand for rail service, the headway of rail lines and the network
development decision change over time. In this dynamic decision making process, the budget
constraint is reformulated to include possible internal funding, such as the fare revenue generated from
the operation of the transit rail lines. The reformulated budget constraint reflects interrelations among
projects in the transit network since the capital used for development is partly supplied by fare revenue
collected from the rail operation. A GA approach is designed to solve the problem, and the properties
of the solution found by the proposed GA are verified.
Results show that (i) the GA approach developed here is capable of finding a quite good and stable
solution at acceptable computaion cost. (ii) The development of the rail transit network can
significantly increase the demand for rail service and reduce the total cost. (iii) Higher travel demand
can encourage more intensive network development and increase the total cost saving. This helps
explain why many large cities in China such as Beijing and Shanghai are investing heavily in transit
development. (iv) A higher budget level can accelerate the development process over the planning
horizon and reduce total costs. The proposed model can serve as a useful tool for making development
plan of transit networks from an economic viability and cost-effectiveness perspective.
Although this paper provides a new venue for addressing the transit network development problem,
some further extensions seem worth pursuing:
1. Travel demand is assumed to be attracted to rail service when OD pairs are connected by rail lines,
but is not affected by the transit service characteristics. However, travelers are usually sensitive to
the travel cost and thus the transit service level (Li et al., 2012a; Peng et al., 2017). Therefore, it
58
seems desirable to extend the proposed model to capture the responses of passengers to the quality
of the rail transit line service.
2. In this paper, the proposed model is deterministic because the demand and supply sides are
assumed to be deterministic. However, in reality there are various random factors (e.g., inflation
and economic changes) which affect the investment of rail lines and the operations of rail services.
It is thus especially important for the authority to consider the investment and operational risks of
rail transit projects in the development issue of urban rail transit networks, which is left for our
future study.
3. This paper focuses mainly on rail mode, and neglects the competition and substitution effects
between private auto and transit modes. It seems desirable to extend the proposed models to
consider different modes and analyze the transit network development in a multi-modal
transportation system (Li et al., 2012b; Ma and Lo, 2013).
4. Urban spatial structure in terms of households’ residential location choices and housing market
has a direct effect on travel demand pattern (Li et al., 2012c; Li and Peng, 2016; Wang and Lo,
2016; Ng and Lo, 2017), and thus on the rail transit service and the network development process.
Therefore, it seems worthwhile to extend the proposed model to explore the effects of urban
spatial structure on transit network development.
Acknowledgments
The work described in this paper was jointly supported by grants from the National Natural Science
Foundation of China (71525003), the Major Program of National Social Science Foundation of China
(13&ZD175), and Huazhong University of Science and Technology (5001300001). The work
conducted at the University of Maryland was partly funded by the University Center for Mobility and
Equity Led by Morgan State University
59
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61
Table 1 Contributions to transportation infrastructure investment models.
Citation Type of
infrastructure
Considering time
horizon or not
Considering interrelation
among projects or not
Wang et al. (2013) Road network × ×
Li et al. (2012) Road network × ×
Luathep et al. (2011) Road network × ×
Gao et al. (2004) Transit network × ×
Sun et al. (2018) Rail line √ ×
Shayanfar et al. (2016) Road network √ √
This paper Transit network √ √
Note: “√” means that the associated item is considered, whereas “×” means that the associated item is not considered.
Table 2 Input data for segments.
Segment
No.
Segment length
(km)
Segment investment costs (million
$)
Associated rail line
1 12 1250 2
2 10 1050 1
3 8 850 2
4 15 1500 1
5 9 950 1
6 18 1800 3
Table 3 Daily travel demands between OD pairs (thousands person trips).
Nodes No.
(O/D) 1 2 3 4 5 6 7
1 0 10 30 11 12 24 25
2 10 0 35 10 27 20 12
3 30 35 0 30 40 25 20
4 11 10 30 0 30 10 15
5 12 27 40 30 0 35 20
6 24 20 25 10 35 0 15
7 25 12 20 15 20 15 0
Table 4 Input parameters for numerical examples.
Symbol Definition Baseline value
1 Value of in-vehicle time ($/h) 15
2 Value of waiting time ($/h) 30
V Average speed of trains (km/h) 40
f Marginal fare by transit ($/km) 0.2
1g Base growth rate of travel demand (year) 0.02
62
2g Annual growth rate caused by network development (year) 0.03
Average number of days of travel per traveler per year 250
Marginal maintenance cost of transit lines (million $/km/year) 5
Annual operating cost per train (million $/year) 3
r Discount rate 0.05
Peak-hour factor 0.1
vehK
Capacity of vehicles (passengers/vehicle) 1500
0p
Selective pressure 0.2
Table 5 Optimized development plan for rail transit network and resulting system performance.
Period
No.
Segment
developed
Completion
time (year)
Train headways of
line 1, 2 and 3 (min)
Daily demand
for rail service
(thousand
person trips)
Discounted
cumulative total
cost saving
(billion $/year) 1h 2h 3h
1 4 6.00 1.54 - - 585.60 7.21
2 6 8.61 1.44 - 2.91 930.27 11.71
3 3 9.47 1.41 2.12 2.86 1209.80 15.25
Notes: (1) The completion time of projects is also the starting or ending time of periods. (2) The discounted
cumulative total cost saving is calculated by the discounted cumulative total cost without investment minus
the that with investment.
Table 6 Comparisons of results obtained by GA and complete enumeration.
GA
(computation time: 0.8 seconds)
Complete enumeration
(computation time: 15 seconds)
Period
No.
Segment
developed
Completion time
(year)
Period
No.
Segment
developed
Completion time
(year)
1 4 6.00 1 4 6.00
2 6 8.61 2 6 8.61
3 3 9.47 3 3 9.47
Table 7 Effects of travel demand on the optimized development plan and system performance.
0.5×base value Base value 1.5×base value
Number of developed projects 2 3 4
Developed projects
(completion time, year)
4 (6.00) 4 (6.00) 4 (6.00)
3 (7.81) 6 (8.61) 6 (7.98)
3 (9.47) 2 (8.76)
3 (9.30)
Total cost saving (billion $) 6.22 15.25 28.62
63
Table 8 Effects of annual budget on the optimized development plan and system performance.
0.8×base value Base value 1.2×base value
Number of developed projects 1 3 4
Developed projects
(completion time)
6 (9.00) 4 (6.00) 4 (5.00)
6 (8.61) 6 (7.46)
3 (9.47) 2 (8.49)
3 (9.20)
Total cost saving (billion $) 0.95 15.25 22.95
Table 9 Input data for segments of Wuhan rail transit network.
Segment
No.
Segment length
(km)
Segment investment costs (million
$)
Associated rail line
1 8.4 2280 1
2 9.8 2325 1
3 3.9 292.5 1
4 10.3 2475 2
5 20.0 3600 2
6 9.2 1380 1
7 6.6 1387.5 1
8 8.0 2400 2
9 12.5 2250 4
10 9.1 1350 2
11 4.6 1275 4
12 10.9 2500 4
13 5.5 990 4
(Sources: http://www.whrt.gov.cn/ and Baidu Map)
Table 10 Initial daily travel demands between OD pairs of Wuhan rail transit network
(thousand person trips).
Nodes No.
(O/D) 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 0 3.2 2.4 6 2.4 3.2 6 3.2 2 1.6 1.6 0.8 0.4 0.4
2 3.2 0 8 10 3.2 4 9.6 3.2 3.2 2.4 2.4 1.6 0.8 0.4
3 2.4 8 0 16 8 8 20 10 4 3.2 3.2 2.4 1.6 0.8
4 6 10 16 0 6.4 10 16 12 8 4.8 4.8 2.4 1.6 1.2
5 2.4 3.2 8 6.4 0 9.6 4.8 4 6.4 4.8 4.8 2.4 1.6 0.8
6 3.2 4 8 10 9.6 0 2.4 2.4 3.2 3.2 3.2 1.6 1.2 0.8
7 6 9.6 20 16 4.8 2.4 0 1.6 3.2 2.4 16 1.6 1.2 0.8
8 3.2 3.2 10 12 4 2.4 1.6 0 2.4 1.6 1.6 1.2 0.8 0.8
9 2 3.2 4 8 6.4 3.2 3.2 2.4 0 4 32 2.4 2.4 1.6
10 1.6 2.4 3.2 4.8 4.8 3.2 2.4 1.6 4 0 8.8 4.8 3.2 2.4
11 1.6 2.4 3.2 4.8 4.8 3.2 16 1.6 32 8.8 0 6.4 1.6 1.2
12 0.8 1.6 2.4 2.4 2.4 1.6 1.6 1.2 2.4 4.8 6.4 0 0.8 0.4
64
13 0.4 0.8 1.6 1.6 1.6 1.2 1.2 0.8 2.4 3.2 1.6 0.8 0 2.4
14 0.4 0.4 0.8 1.2 0.8 0.8 0.8 0.8 1.6 2.4 1.2 0.4 2.4 0
Table 11 Optimized network development plan and headways of rail lines in Wuhan.
Period
No.
Segment
developed
Completion
time (year)
Train headways of
line 1, 2 and 4 (min)
Daily demand
for rail service
(thousand
person trips)
Discounted
cumulative total
cost
(billion $/year) 1h 2h 3h
1 3 0.29 1.38 - - 292.89 19.04
2 6 1.60 1.59 - - 564.44 38.78
3 8 3.63 1.51 2.09 - 862.10 47.77
4 10 4.67 1.46 1.94 - 1079.08 60.89
5 4 6.45 1.38 1.28 - 1353.40 70.51
6 2 7.98 1.04 1.22 - 1608.85 82.04
7 5 10.17 0.96 0.97 - 1869.98 87.57
8 9 11.40 0.91 0.92 3.90 2080.53 90.22
9 11 12.06 0.89 0.90 3.07 2277.75 92.85
10 7 12.76 0.78 0.87 3.03 2460.48 96.71
11 1 13.86 0.69 0.83 2.96 2672.02 99.28
65
Fig.1. Development process of rail transit network in Wuhan, China
(Sources: http://www.whrt.gov.cn/ and https://en.wikipedia.org/wiki/Wuhan_Metro).
Fig. 2. Example of a chromosome.
66
Fig. 3. Example of an urban rail transit network.
(a) t=1(year 6.00- 8.61) (b) t=2 (year 8.61- 9.47) (c) t=3 (year 9.47- 10)
Fig. 4. Evolution of the state of the rail transit network
2 3 5 7
1
4
6
Line 1
Line 2
Line 31
42 5
3
6
2 3 5 7
1
4
6
Line 1
Line 2
Line 31
42 5
3
6
2 3 5 7
1
4
6
Line 1
Line 2
Line 31
42 5
3
6
2 3 5 7
1
4
6
Line 1
Line 2
Line 31
42 5
3
6
67
Fig. 5. Changes of discounted cumulative total cost with and without the rail investment.
(a)
71.75
93.90
99.54
102.28
70.63
101.11
111.24
117.53
70.00
75.00
80.00
85.00
90.00
95.00
100.00
105.00
110.00
115.00
120.00
6.00 8.61 9.47 10
Dis
cou
nte
d c
um
ula
tive
tota
l co
st (
bil
lion
$)
Invest time (year)
With investment
Without investment
68
(b)
Fig. 6. Map of Wuhan subway lines (blue for Line 1, purple for Line 2 and green for Line 4): (a) urban
rail transit network of Wuhan, China; (b) candidate rail transit projects.
Fig. 7. Fitted generalized extreme value distribution of the fitness values of the sample.
